Throughout, \(F\) will denote either the real or complex numbers.
We began class by looking at examples of vector spaces, initially, the vector space \(\mathbb{R}^3\) of column vectors defined over the real numbers. Beginning with the basic properties of vector addition, where for \(v_1 = \begin{pmatrix} \alpha_1\\\beta_1\\\gamma_1 \end{pmatrix}\) and \(v_2 = \begin{pmatrix}\alpha_2\\\beta_2\\\gamma_2\end{pmatrix}\), \(v_1+v_2 := \begin{pmatrix}\alpha_1+\alpha_2\\\beta_1+\beta_2\\\gamma_1+\gamma_2\end{pmatrix}\), and scalar multiplication, \(\lambda v_1 := \begin{pmatrix} \lambda\alpha_1\\\lambda\beta_1\\\lambda\gamma_1\end{pmatrix}\), we discussed the following properties (and verified a few of them), all which follow from similar familiar properties of \(\mathbb{R}\):
- (i) The zero vector \(\vec{0} = \begin{pmatrix}0\\0\\0\end{pmatrix}\) has the property that \(\vec{0}+v = v\), for all \(v\in \mathbb{R}^3\). (Existence of additive identity)
- (ii) For \(v = \begin{pmatrix}\alpha\\\beta\\\gamma\end{pmatrix}\), \(-v+v = \vec{0}\), where \(-v := \begin{pmatrix}-\alpha\\-\beta\\-\gamma\end{pmatrix}\). (Existence of additive inverses)
- (iii) \(v_1+v_2 = v_2+v_1\), for all \(v_1, v_2\in \mathbb{R}^3\). (Commutativity of addition)
- (iv) \(v_1+(v_2+v_3) = (v_1+v_2)+v_3\), for all \(v_i\in \mathbb{R}^3\). (Associativity of addition)
- (v) \(\lambda(v_1+v_2) = \lambda v_1+\lambda v_2\), for all \(\lambda\in \mathbb{R}\) and \(v_i\in \mathbb{R}^3\). (First distributive property)
- (vi) \((\lambda+\gamma)v = \lambda v+\gamma v\), for all \(\lambda, \gamma\in \mathbb{R}\) and \(v\in \mathbb{R}^3\). (Second distributive property)
- (vii) \((\lambda\gamma)v = \lambda(\gamma v)\), for all \(\lambda, \gamma\in \mathbb{R}\) and \(v\in \mathbb{R}^3\). (Associativity of scalar multiplication)
- (viii) \(1\cdot v = v\), for all \(v\in \mathbb{R}^3\).
We then looked at the vector space \(P(2)\) of polynomials of degree two or less over \(\mathbb{R}\) and noted that since a typical element in \(P(2)\) has the form \(\alpha+\beta x+\gamma x^2\), when we add two expressions of this form, or multiply them by a scalar, the resulting expressions look very similar to what we get when we add or scalar multiply vectors in \(\mathbb{R}^3\). Something similar happens, if, for example, we take three vectors \(u, v, w\in \mathbb{R}^{17}\) and consider all expressions of the form \(\alpha u+\beta v+\gamma w\). This gives a vector space that looks very similar to \(\mathbb{R}^3\) and \(P(2)\). These examples show the advantage of defining vector spaces in an abstract setting in a way that captures all of the properties of particular vector spaces we might encounter in different contexts.
We also noted that \(\mathbb{R}^n\) and \(\mathrm{M}_2(\mathbb{R})\), the set of \(2\times 2\) matrices over \(\mathbb{R}\), form vector spaces over \(\mathbb{R}\) and \(\mathbb{C}^n\), with coordinate-wise addition and scalar multiplication, is a vector space over \(\mathbb{C}\). This led to the following:
Definition. Let \(F\) denote either \(\mathbb{R}\) or \(\mathbb{C}\). A vector space over \(F\) is a set \(V\) together with two operations, addition of elements of \(V\) and multiplication of elements from \(F\) times elements in \(V\), satisfying the eight properties above:
- (i) There exists a zero vector \(\vec{0}\in V\) satisfying \(v+\vec{0} = v\), for all \(v\in V\). (Existence of additive identity)
- (ii) For each \(v\in V\), there exists \(-v\in V\) such that \(v+(-v) = \vec{0}\). (Existence of additive inverses)
- (iii) \(v_1+v_2 = v_2+v_1\), for all \(v_1, v_2\in V\). (Commutativity of addition)
- (iv) \(v_1+(v_2+v_3) = (v_1+v_2)+v_3\), for all \(v_i\in V\). (Associativity of addition)
- (v) \(\lambda(v_1+v_2) = \lambda v_1+\lambda v_2\), for all \(\lambda\in F\) and \(v_i\in V\). (First distributive property)
- (vi) \((\lambda+\gamma)v = \lambda v+\gamma v\), for all \(\lambda, \gamma\in F\) and \(v\in V\). (Second distributive property)
- (vii) \((\lambda\gamma)v = \lambda(\gamma v)\), for all \(\lambda, \gamma\in F\) and \(v\in V\). (Associativity of scalar multiplication)
- (viii) \(1\cdot v = v\), for all \(v\in V\).
We ended class by recording the following proposition and proving the first two cases.
Proposition. Let \(V\) be a vector space over \(F\). The following properties hold:
- (i) Cancellation holds: For all \(u, v, w\in V\), if \(v+w = v+u\), then \(w = u\).
- (ii) The additive identity \(\vec{0}\) is unique.
- (iii) \(0\cdot v = \vec{0}\), for all \(v\in V\).
- (iv) For any \(v\in V\), its additive inverse \(-v\) is unique.
- (v) For all \(\lambda\in F\) and \(v\in V\), \(-\lambda\cdot v = -(\lambda v)\). In particular, \(-1\cdot v = -v\), for all \(v\in V\).
We began class by reviewing the axioms for a vector space \(V\) over \(F\) established in the previous lecture. We also gave proofs of properties (iv) and (v) from the proposition at the end of the previous lecture. The rest of the class we spent discussing the following definition and examples thereof.
Definition. Let \(V\) be a vector space over \(F\). A subset \(W\subseteq V\) is a subspace of \(V\) if it satisfies:
- (i) \(W\) is closed under addition, i.e., if \(w_1, w_2\in W\), then \(w_1+w_2\in W\).
- (ii) \(W\) is closed under scalar multiplication, i.e., if \(\lambda\in F\) and \(w\in W\), then \(\lambda w\in W\).
We noted that it follows that \(\vec{0}\in W\) and \(-w\in W\), whenever \(w\in W\), so that \(W\) is a vector space in its own right under the given operations on \(V\). We then discussed the following examples.
Subspaces of \(\mathbb{R}^2\). We saw that the following are subspaces of \(\mathbb{R}^2\): \(\{\vec{0}\}\), lines through the origin, and \(\mathbb{R}^2\), the latter being an improper subspace. We noted, but did not prove, that these are the only subspaces of \(\mathbb{R}^2\).
Subspaces of \(\mathbb{R}^3\). We saw that the following are subspaces of \(\mathbb{R}^3\): \(\{\vec{0}\}\), lines through the origin, planes through the origin, and \(\mathbb{R}^3\). We noted, but did not prove, that these are the only subspaces of \(\mathbb{R}^3\).
Solutions to systems of equations. We saw that the set of solutions to a homogeneous system of linear equations in \(n\) unknowns form a subspace of \(\mathbb{R}^n\).
Eigenspaces. Given an \(n\times n\) matrix over \(\mathbb{R}\) and \(\lambda\in\mathbb{R}\), we saw that the set of \(v\in\mathbb{R}^n\) satisfying \(Av = \lambda v\) forms a subspace of \(\mathbb{R}^n\).
The span of a set of vectors. Given a finite set of vectors \(v_1, \ldots, v_t\) in the vector space \(V\), we defined \(\mathrm{Span}\{v_1, \ldots, v_t\}\) to be the set of all \(F\)-linear combinations of \(v_1, \ldots, v_t\), i.e., the set of all expressions of the form \(\alpha_1 v_1+\cdots+\alpha_t v_t\), with each \(\alpha_i\in F\), and noted that \(\mathrm{Span}\{v_1, \ldots, v_t\}\) is a subspace of \(V\). We call this subspace the subspace of \(V\) spanned by \(v_1, \ldots, v_t\).
We ended class by observing that the unit vectors \(e_1, e_2, e_3\in\mathbb{R}^3\) span \(\mathbb{R}^3\).
We began class by reviewing the definition of subspace of a vector space and then showed that the set \(\{A\in\mathrm{M}_2(\mathbb{R})\ |\ \mathrm{tr}(A) = 0\}\) is a subspace of \(\mathrm{M}_2(\mathbb{R})\), where \(\mathrm{tr}(A)\) denotes the trace of the matrix \(A\), which, if \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\), is given by \(\mathrm{tr}(A) = a+d\).
For subspaces \(W_1, W_2\) contained in the vector space \(V\), we then considered two constructions:
Definition. The sets \(W_1+W_2\) and \(W_1\cap W_2\), the sum of \(W_1, W_2\) and the intersection of \(W_1, W_2\), are defined as follows:
- (i) \(W_1+W_2 := \{w_1+w_2\ |\ w_1\in W_1\ \text{and}\ w_2\in W_2\}\).
- (ii) \(W_1\cap W_2 = \{v\ |\ v\in W_1\ \text{and}\ v\in W_2\}\).
We showed that \(W_1+W_2\) and \(W_1\cap W_2\) are subspaces of \(V\). We also noted that:
- (i) \(\mathbb{R}^2 = W_1+W_2\), where \(W_1\) and \(W_2\) are two distinct lines through the origin in \(\mathbb{R}^2\), and
- (ii) \(\mathrm{M}_2(\mathbb{R}) = W_1+W_2\), where \(W_1\) consists of all matrices of the form \(\begin{pmatrix} a & 0\\0 & d\end{pmatrix}\) and \(W_2\) consists of all matrices of the form \(\begin{pmatrix} 0 & b\\c & 0\end{pmatrix}\). We noted that for these examples, \(W_1\cap W_2 = \{\vec{0}\}\), which turns out to be an important property.
We then discussed how \(\mathbb{R}^3\) is the sum of two subspaces that are distinct planes through the origin, with non-zero intersection. For this, we appealed to our geometric intuition and our basic knowledge of linear algebra, namely, the points in the intersection of two such planes corresponds to the solution set of a homogeneous system of two equations in three variables, which always has a one parameter solution set.
This was followed by a brief discussion about subspaces spanned by a set of vectors, noting in particular that
We ended class with a discussion of the following definition.
Thus \(\mathbb{R}^2\) is the direct sum of any two distinct lines through the origin and \(\mathrm{M}_2(\mathbb{R})\) is the direct sum of the space of matrices of the form \(\begin{pmatrix} a & 0\\0 & d\end{pmatrix}\) with the space of matrices of the form \(\begin{pmatrix} 0 & b\\c & 0\end{pmatrix}\).
The first fifteen minutes of class were devoted to Quiz 1. We then reviewed the definition of direct sum and worked through the following proposition.
We then discussed at length how to determine if a column vector belongs to the span of a given set of column vectors, illustrating this with the question of when does \(v = \begin{pmatrix} a\\b\\c\\d\end{pmatrix}\) belong to the span of the vectors \(A = \begin{pmatrix} v_1\\v_2\\v_3\\v_4\end{pmatrix}\), \(B = \begin{pmatrix} u_1\\u_2\\u_3\\u_4\end{pmatrix}\), \(C = \begin{pmatrix} w_1\\w_2\\w_3\\w_4\end{pmatrix}\). We noted that \(v\in\mathrm{Span}\{A, B, C\}\) if and only if \(v\) is a solution to the system of linear equations
We then reviewed at length how to solve such a system of equations via Gaussian elimination: One forms the augmented matrix \(\left[\begin{array}{ccc|c} v_1 & u_1 & w_1 & a\\v_2 & u_2 & w_2 & b\\v_3 & u_3 & w_3 & c\\v_4 & u_4 & w_4 & d\end{array}\right]\) and uses elementary row operations to put the augmented matrix into RREF, i.e., reduced row echelon form.
Reduced Row Echelon Form. Suppose \(L\) is an augmented matrix arising from a system of linear equations over \(F\). Then \(L\) is in reduced row echelon form if it satisfies the following conditions:
- (i) The first entry of any non-zero row is 1. This is called a leading 1 or pivot.
- (ii) \(L\) has zeros above and below each leading 1.
- (iii) The leading 1s move from left to right as one moves down the rows of \(L\).
- (iv) All non-zero rows are at the top of the matrix.
We then discussed the algorithm for rendering an augmented matrix into its RREF, whereby the solutions to the system can be read off. We noted the crucial:
We continued our discussion of spanning properties by recording a general statement analogous to the one presented in the previous lecture, namely, given vectors \(w = \begin{pmatrix} a_1\\\vdots\\a_m\end{pmatrix}\), \(v_1, \ldots, v_n\) in \(\mathbb{R}^m\), with \(v_j = \begin{pmatrix} \alpha_{1j}\\\vdots\\\alpha_{mj}\end{pmatrix}\), then \(w\in\mathrm{Span}\{v_1, \ldots, v_n\}\) if and only if the system of equations
has a solution.
We then made the following crucial observation which plays a key role throughout this semester.
We then discussed at length the following crucial definitions.
Definitions. Let \(V\) be a vector space over \(F\) and \(v_1, \ldots, v_r\in V\).
- (i) \(v_1, \ldots, v_r\) are linearly dependent if some \(v_j\) is in the span of the remaining vectors, in other words, \(v_j\in\mathrm{Span}\{v_1, \ldots, \widehat{v_j}, \ldots, v_r\}\).
- (ii) \(v_1, \ldots, v_r\) are linearly independent if they are not linearly dependent, i.e., no \(v_j\) is in the span of the remaining vectors.
These definitions were re-formulated in the following observation.
The observation had the immediate consequence:
Returning to the setting of column vectors in \(\mathbb{R}^m\), using the crucial observation, we summarized our discussion as follows.
Summary. Suppose we have vectors \(w = \begin{pmatrix} a_1\\\vdots\\a_m\end{pmatrix}\), \(v_1, \ldots, v_n\) in \(\mathbb{R}^m\), with \(v_j = \begin{pmatrix}\alpha_{1j}\\\vdots\\\alpha_{mj}\end{pmatrix}\). Let \(A\) denote the \(m\times n\) matrix whose columns are \(v_1, \ldots, v_n\). Then, upon setting \(X = \begin{pmatrix} x_1\\\vdots\\x_n\end{pmatrix}\), we have:
- (i) \(w\in\mathrm{Span}\{v_1, \ldots, v_n\}\) if and only if the system of equations \(AX = w\) has a solution.
- (ii) The vectors \(v_1, \ldots, v_n\) are linearly dependent if and only if the system of equations \(AX = \vec{0}\) has a non-zero solution.
- (iii) The vectors \(v_1, \ldots, v_n\) are linearly independent if and only if the system of equations \(AX = \vec{0}\) has only the zero solution.
We ended class by noting that a vector space is finite dimensional if it is the span of a finite set of vectors. We also discussed informally how a redundant spanning set can be reduced to a set that spans efficiently. In particular, we showed that if \(V = \mathrm{Span}\{v_1, v_2, v_3\}\) and \(v_2\in\mathrm{Span}\{v_1, v_3\}\), then \(V = \mathrm{Span}\{v_1, v_3\}\).
We began class by reviewing the definitions of linear dependence and linear independence as well as the equivalent conditions for each stated in the previous lecture. We then used Gaussian elimination to show that the vectors \(v_1 = \begin{pmatrix} 1\\1\\0\\0\end{pmatrix}\), \(v_2 = \begin{pmatrix} 0\\1\\1\\0\end{pmatrix}\), \(v_3 = \begin{pmatrix} 0\\0\\1\\1\end{pmatrix}\) are linearly independent and that \(w = \begin{pmatrix} 1\\0\\2\\3\end{pmatrix}\in\mathrm{Span}\{v_1, v_2, v_3\}\).
Most of the remainder of the class was spent discussing at length the Fundamental Property: In any vector space, the number of elements in any spanning set is always greater than or equal to the number of elements in a set of linearly independent vectors. We then worked through a special case of the
Exchange Theorem. Let \(V\) be a vector space with \(V = \mathrm{Span}\{v_1, \ldots, v_r\}\) and assume \(w_1, \ldots, w_s\in V\) are linearly independent in \(V\). Then \(s\leq r\) and after re-indexing the \(v_j\)'s, we have \(V = \mathrm{Span}\{w_1, \ldots, w_s, v_{s+1}, \ldots, v_r\}\).
In particular, we showed explicitly that one cannot have \(V = \mathrm{Span}\{v_1, v_2\}\) and \(w_1, w_2, w_3\) linearly independent in \(V\). This process involved exchanging \(w_1, w_2\) for \(v_1, v_2\). A key observation was that after each exchange, we had a new set of vectors that still spanned \(V\).
We ended class with the following corollary and its consequences.
Definitions.
- (i) Let \(B := \{v_1, \ldots, v_n\}\subseteq V\) be a linearly independent set such that \(V = \mathrm{Span}\{v_1, \ldots, v_n\}\). Then \(B\) is called a basis for \(V\). Note that by the previous corollary, any two bases have the same number of elements.
- (ii) The dimension of \(V\) is the number of elements in any basis of \(V\). We often write \(\dim(V) = n\), if \(V\) has a basis with \(n\) elements.
The first thirteen minutes of class were devoted to Quiz 2. We then reviewed the Exchange Theorem and the concepts of basis and dimension for an arbitrary vector space. We then gave the following examples.
Examples.
- (i) The vectors \(e_1 = \begin{pmatrix} 1\\0\\\vdots\\0\\0\end{pmatrix}, e_2 = \begin{pmatrix} 0\\1\\0\\\vdots\\0\end{pmatrix}, \ldots, e_n = \begin{pmatrix} 0\\0\\\vdots\\0\\1\end{pmatrix}\) form a basis for \(\mathbb{R}^n\) or \(\mathbb{C}^n\). Thus, these two vector spaces have dimension \(n\).
- (ii) \(\mathbb{C}\) can be regarded as a vector space over \(\mathbb{R}\), in which case, \(1, i\) forms a basis for \(\mathbb{C}\) over \(\mathbb{R}\). Thus, \(\dim_{\mathbb{R}}(\mathbb{C}) = 2\).
- (iii) The polynomials \(1, x, \ldots, x^n\) form a basis for the space of polynomials over \(F\) having degree less than or equal to \(n\). Thus, this vector space has dimension \(n+1\).
- (iv) Let \(E_{ij}\) denote the \(m\times n\) matrix over \(F\) having 1 in its \((i,j)\) entry and 0 in all other entries. Then \(\{E_{ij}\}\) forms a basis for the space of \(m\times n\) matrices over \(F\). Thus, this vector space has dimension \(mn\).
We then discussed the following proposition.
Proposition. Let \(V\) be a vector space of dimension \(n\) over \(F\).
- (i) If the set \(S = \{v_1, \ldots, v_m\}\) spans \(V\), then some subset of \(S\) forms a basis for \(V\).
- (ii) If the set \(T = \{u_1, \ldots, u_r\}\) is linearly independent, then \(T\) can be extended to a basis for \(V\).
We began class by reviewing the statement of the proposition given at the end of the previous lecture. We then discussed at length the following theorem.
Theorem. Let \(V\) be a vector space with \(\dim(V) = n\) and \(v_1, \ldots, v_n\in V\). Then the following are equivalent:
- (i) \(v_1, \ldots, v_n\) form a basis for \(V\).
- (ii) \(\mathrm{Span}\{v_1, \ldots, v_n\} = V\).
- (iii) \(v_1, \ldots, v_n\) are linearly independent.
The crucial point about the theorem above is that the dimension of \(V\) is given a priori to be \(n\). The theorem above was followed by a discussion of the following special case involving column vectors in \(\mathbb{R}^n\) or \(\mathbb{C}^n\).
Very Important Special Case of Column Vectors. Suppose \(v_1, \ldots, v_n\) are column vectors in \(\mathbb{R}^n\) or \(\mathbb{C}^n\) and let \(A\) denote the \(n\times n\) matrix whose columns are \(v_1, \ldots, v_n\). Then the following conditions are equivalent.
- (i) \(v_1, \ldots, v_n\) form a basis for \(\mathbb{R}^n\) or \(\mathbb{C}^n\).
- (ii) \(\mathrm{Span}\{v_1, \ldots, v_n\} = \mathbb{R}^n\) or \(\mathbb{C}^n\).
- (iii) \(v_1, \ldots, v_n\) are linearly independent.
- (iv) The system of equations \(AX = \vec{0}\) has only the zero solution, where \(X = \begin{pmatrix} x_1\\x_2\\\vdots\\x_n\end{pmatrix}\).
- (v) The system of equations \(AX = b\) has a (unique) solution for any \(b\in\mathbb{R}^n\).
- (vi) The RREF of \(A\) equals \(I_n\), the \(n\times n\) identity matrix.
- (vii) \(A\) is invertible, i.e., there exists an \(n\times n\) matrix \(A^{-1}\) such that \(A^{-1}A = I_n = AA^{-1}\).
- (viii) \(\det A\), the determinant of \(A\), is not zero.
As a corollary to the discussion above, we noted:
Uniqueness of Representation. If \(v_1, \ldots, v_n\) is a basis for \(V\), then every \(v\in V\) can be written uniquely as a linear combination of \(v_1, \ldots, v_n\). In other words, if \(v\in V\) and
then \(\alpha_1 = \beta_1, \ldots, \alpha_n = \beta_n\).
We then turned to a concept that plays a central role in this course.
Definition. Let \(T: V\to W\) be a function between the \(F\)-vector spaces \(V\) and \(W\). Then \(T\) is a linear transformation if it satisfies:
- (i) \(T(v_1+v_2) = T(v_1)+T(v_2)\), for all \(v_1, v_2\in V\).
- (ii) \(T(\lambda v) = \lambda T(v)\), for all \(\lambda\in F\) and \(v\in V\).
We ended class with the following examples of linear transformations.
Examples.
- (i) Let \(A\) denote an \(m\times n\) matrix over \(F\). For a column vector \(v\in F^n\), define \(T_A(v) := A\cdot v\in F^m\). Then \(T_A\) is a linear transformation.
- (ii) Rotation of \(\mathbb{R}^2\) counter-clockwise through an angle of \(\theta\) is a linear transformation from \(\mathbb{R}^2\) to \(\mathbb{R}^2\). This can be seen as follows: the transformation \(T\begin{pmatrix} a\\b\end{pmatrix} = \begin{pmatrix} a\cos(\theta)-b\sin(\theta)\\a\sin(\theta)+b\cos(\theta)\end{pmatrix}\) rotates the vector \(\begin{pmatrix} a\\b\end{pmatrix}\) by \(\theta\) degrees counter-clockwise and is given by multiplication by \(A = \begin{pmatrix} \cos(\theta) & -\sin(\theta)\\\sin(\theta) & \cos(\theta)\end{pmatrix}\), so that \(T = T_A\). Note that the columns of \(A\) are orthogonal and have length one. In this case \(A\) is called an orthogonal matrix, a property to be considered later in the semester.
- (iii) Let \(V\) be the vector space of polynomials over \(\mathbb{R}\) (or \(\mathbb{C}\)) having degree less than or equal to \(n\). Define \(T: V\to V\) by \(T(p(x)) = p'(x)\), the derivative of \(p(x)\). Then \(T\) is a linear transformation.
The first 13 minutes of class were devoted to Quiz 3. We then spent most of the remaining time in class discussing the following algorithm.
Algorithm for finding a basis. Suppose \(W\) is a subspace of \(\mathbb{R}^m\) or \(\mathbb{C}^m\) spanned by the column vectors \(v_1, \ldots, v_n\) and let \(A\) denote the \(m\times n\) matrix whose columns are \(v_1, \ldots, v_n\). Let \(R\) denote the RREF form of \(A\) and suppose the columns of \(R\) are \(w_1, \ldots, w_n\). Let \(w_{i_1}, \ldots, w_{i_r}\) denote the columns of \(R\) containing the pivots of \(R\), i.e., those columns consisting of one 1 and having all remaining entries equal to zero. Then \(v_{i_1}, \ldots, v_{i_r}\) form a basis for \(W\).
We illustrated this algorithm on the matrix \(A = \begin{pmatrix} 1 & 2 & 3 & 0\\1 & 2 & 3 & 0\\0 & 3 & 4 & 1\\0 & 3 & 4 & 1\end{pmatrix}\). We also noted that the algorithm follows from the two facts: (i) \(w_{i_1}, \ldots, w_{i_r}\) is clearly a basis for the column space of \(R\) and (ii) the solution spaces associated to \(A\) and \(R\) are the same, and therefore any dependence relation on \(v_1, \ldots, v_n\) is a dependence relation on \(w_1, \ldots, w_n\) and conversely. We then showed why elementary row operations preserve solution spaces.
We ended class by returning to our discussion of linear transformations and noted the following.
Two properties of linear transformations. Let \(T: V\to W\) be a linear transformation between vector spaces over \(F\). Then:
- (i) \(T(\vec{0}_V) = \vec{0}_W\).
- (ii) \(T(-v) = -T(v)\), for all \(v\in V\).
We began class with the following set up. \(T: V\to W\) is a linear transformation of vector spaces over \(F\), \(\alpha = \{v_1, \ldots, v_n\}\) is a basis for \(V\) and \(\beta = \{w_1, \ldots, w_m\}\) is a basis for \(W\). We then discussed the following.
Important points about linear transformations. In the notation established above,
- (i) \(T\) is determined by its effect on a basis. In particular, given values \(T(v_1) = w_1, \ldots, T(v_n) = w_n\), we can calculate the value \(T(v)\) for any \(v\in V\), namely, if \(v = a_1v_1+\cdots+a_nv_n\), then we have \(T(v) = a_1w_1+\cdots+a_nw_n\).
- (ii) To define a linear transformation \(T:V\to W\), it suffices to assign values to the basis elements \(v_1, \ldots, v_n\). The value of an arbitrary \(v\in V\) is then determined as follows: If \(T(v_i) = w_i\), for \(i = 1,\ldots, n\) and \(v = b_1v_1+\cdots+b_nv_n\), then \(T(v) = b_1w_1+\cdots+b_nw_m\).
We then gave the following very important definition, using the notation established above.
We then worked the following examples.
Examples. 1. Let \(T: \mathbb{R}^2\to\mathbb{R}^2\) be defined by \(T\begin{pmatrix} x\\y\end{pmatrix} = \begin{pmatrix} 3x-2y\\x+4y\end{pmatrix}\), and write \(e_1, e_2\) for the standard basis of \(\mathbb{R}^2\).
- (i) For \(\alpha = \{e_1, e_2\} = \beta\), we saw that \([T]_{\alpha}^{\beta} = \begin{pmatrix} 3 & -2\\1 & 4\end{pmatrix}\).
- (ii) For \(\alpha' = \left\{\begin{pmatrix} 1\\1\end{pmatrix}, \begin{pmatrix} -1\\1\end{pmatrix}\right\}\), we saw that \([T]_{\alpha'}^{\beta} = \begin{pmatrix} 1 & -5\\5 & 3\end{pmatrix}\).
- (iii) For \(\alpha'\) as in (ii) and \(\beta' = \left\{\begin{pmatrix} 1\\2\end{pmatrix}, \begin{pmatrix} 1\\1\end{pmatrix}\right\}\), we saw that \([T]_{\alpha'}^{\beta'} = \begin{pmatrix} 4 & 8\\-3 & -13\end{pmatrix}\).
2. Taking \(V\) to be the space of real polynomials of degree less than or equal to three, with basis \(\alpha = \{1, x, x^2, x^3\} = \beta\), and \(T(p(x)) = p'(x)\), we saw that