- #1

- 56

- 3

## Homework Statement

Find the dimension of the subspace of all vectors in ##\mathbb{R}^3## whose first and third entries are equal.

## Homework Equations

## The Attempt at a Solution

So I arrived at two solutions and I'm not entirely sure which is the valid one.

#1

Let ##H \text{ be a subspace of } \mathbb{R}^3##

##H = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \mid a, b, c \in \mathbb{R}^3, a = c \right\} ##

## a \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + b \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + c \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \rightarrow \mathcal{B} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\} ##

Dim = # elements in ##\mathcal{B}## = 3

#2

Let ##H \text{ be a subspace of } \mathbb{R}^3##

## H = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \mid a, b \in \mathbb{R}^3 \right\} ##

## a \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \rightarrow \mathcal{B} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right\} ##

Dim = # elements in ##\mathcal{B}## = 2

I believe that the first one is correct, or at least more correct if it's wrong, since to span ##\mathbb{R}^3## we would need at 3 vectors.