Fall 2026: Math 590 Quiz 1 Solutions

Problem 1

Let \(V\) denote the set \(\mathbb{R}_+^2\) with the following binary operations: \((x_1, y_1)+'(x_2, y_2) := (x_1x_2, y_1y_2)\) and \(c\cdot'(x, y) := (x^c, y^c)\). Verify the following vector space axioms, where the scalars are from \(\mathbb{R}\):

  1. (i) Additive inverses exist.
  2. (ii) Scalar multiplication distributes over vector sum.
  3. (iii) Scalar multiplication is associative.

Solution. (i) The additive identity for \(+'\) is \((1,1)\), since

\[(x,y)+'(1,1) = (x\cdot 1,\, y\cdot 1) = (x,y).\]

Given \((x,y)\in V\), take \(\left(\frac{1}{x}, \frac{1}{y}\right)\in V\) (this lies in \(V\) since \(x, y > 0\) implies \(1/x, 1/y > 0\)). Then

\[(x,y)+'\left(\frac{1}{x}, \frac{1}{y}\right) = \left(x\cdot\frac{1}{x},\ y\cdot\frac{1}{y}\right) = (1,1),\]

so every element of \(V\) has an additive inverse with respect to \(+'\).

(ii) Let \(c\in\mathbb{R}\) and \((x_1, y_1), (x_2, y_2)\in V\). Then

\[c\cdot'\big((x_1,y_1)+'(x_2,y_2)\big) = c\cdot'(x_1x_2,\, y_1y_2) = \big((x_1x_2)^c,\ (y_1y_2)^c\big) = \big(x_1^cx_2^c,\ y_1^cy_2^c\big).\]

On the other hand,

\[c\cdot'(x_1,y_1)+'c\cdot'(x_2,y_2) = (x_1^c, y_1^c)+'(x_2^c, y_2^c) = (x_1^cx_2^c,\ y_1^cy_2^c).\]

The two expressions agree, so \(c\cdot'\big((x_1,y_1)+'(x_2,y_2)\big) = c\cdot'(x_1,y_1)+'c\cdot'(x_2,y_2)\).

(iii) Let \(c, d\in\mathbb{R}\) and \((x,y)\in V\). Then

\[c\cdot'\big(d\cdot'(x,y)\big) = c\cdot'(x^d, y^d) = \big((x^d)^c,\ (y^d)^c\big) = (x^{cd}, y^{cd}).\]

On the other hand,

\[(cd)\cdot'(x,y) = (x^{cd}, y^{cd}).\]

Since these agree, \(c\cdot'(d\cdot'(x,y)) = (cd)\cdot'(x,y)\).

Problem 2

Let \(V\) be a vector space over \(F\) and \(W_1, W_2\) subspaces of \(V\). First define what it means for \(V\) to be the direct sum of \(W_1\) and \(W_2\) and then give an example of spaces \(V, W_1, W_2\) such that \(V = W_1\bigoplus W_2\).

Solution. \(V\) is the (internal) direct sum of \(W_1\) and \(W_2\), written \(V = W_1\bigoplus W_2\), if both of the following hold:

  1. (a) \(V = W_1+W_2\), i.e., every \(v\in V\) can be written as \(v = w_1+w_2\) for some \(w_1\in W_1\), \(w_2\in W_2\); and
  2. (b) \(W_1\cap W_2 = \{0\}\).

Example. Let \(V = \mathbb{R}^2\), and let

\[W_1 = \{(x,0) : x\in\mathbb{R}\}, \qquad W_2 = \{(0,y) : y\in\mathbb{R}\}\]

be the \(x\)-axis and \(y\)-axis, respectively. Each \(W_i\) is a subspace of \(V\). Given any \((x,y)\in V\), we have \((x,y) = (x,0)+(0,y)\) with \((x,0)\in W_1\) and \((0,y)\in W_2\), so \(V = W_1+W_2\). Moreover if \((x,0) = (0,y)\) then \(x = 0\) and \(y = 0\), so \(W_1\cap W_2 = \{(0,0)\}\). Hence \(V = W_1\bigoplus W_2\).