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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