## Linear Algebra and Its Applications, Exercise 2.3.13

Exercise 2.3.13.What are the dimensions of the following spaces?

a) vectors in $\mathbf{R}^4$ with components that sum to zero

b) the nullspace associated with the 4 by 4 identity matrix

c) the space of all 4 by 4 matrices

Answer: a) If the components of a vector $v$ sum to zero then we have $v_1 + v_2 + v_3 + v_4 = 0 \rightarrow v_4 = -v_1 - v_2 - v_3$

Such a vector $v$ can be expressed as the linear combination of three vectors as follows: $v = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ -v_1 - v_2 - v_3 \end{bmatrix} = v_1 \begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \end{bmatrix} + v_2 \begin{bmatrix} 0 \\ 1 \\ 0 \\ -1 \end{bmatrix} + v_3 \begin{bmatrix} 0 \\ 0 \\ 1 \\ -1 \end{bmatrix}$

The three vectors are linearly independent (as can be easily seen by doing elimination on a matrix whose columns are the vectors) and since they span the space they are a basis for it. The dimension of the space is therefore 3.

b) The nullspace of the 4 by 4 identity matrix $I$ contains solutions to $Ix = 0$ or $\begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

This is equivalent to $x_1 \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

But since these vectors are linearly independent the only solution is $x_1 = x_2 = x_3 = x_4 = 0$ or $x = (0, 0, 0, 0)$. The nullspace consists only of the zero vector and (by convention) its dimension is zero.

c) The space of all 4 by 4 matrices has as a basis the set of matrices in which one element is one and the rest are zero; for example $\begin{bmatrix} 1&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix} \quad \begin{bmatrix} 0&1&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix}$ $\cdots \quad \begin{bmatrix} 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&1&0 \end{bmatrix} \quad \begin{bmatrix} 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&1 \end{bmatrix}$

There are 16 such matrixes in this basis, so the dimension of the space is 16.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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