Linear Algebra and Its Applications, Review Exercise 2.4

Review exercise 2.4. Given the matrix

A = \begin{bmatrix} 1&2&0&2&1 \\ -1&-2&1&1&0 \\ 1&2&-3&-7&-2\end{bmatrix}

find its echelon form U and the dimensions of the column space, nullspace, row space, and left nullspace of A.

Answer: We perform elimination on A to reduce it to echelon form. In the first step we multiply the first row of A by -1 and subtract it from the second row:

\begin{bmatrix} 1&2&0&2&1 \\ -1&-2&1&1&0 \\ 1&2&-3&-7&-2\end{bmatrix}

\Rightarrow \begin{bmatrix} 1&2&0&2&1 \\ 0&0&1&3&1 \\ 1&2&-3&-7&-2\end{bmatrix}

and then multiply the first row of A by 1 and subtract it from the third row:

\begin{bmatrix} 1&2&0&2&1 \\ 0&0&1&3&1 \\ 1&2&-3&-7&-2\end{bmatrix}

\Rightarrow \begin{bmatrix} 1&2&0&2&1 \\ 0&0&1&3&1 \\ 0&0&-3&-9&-3\end{bmatrix}

Finally we multiply the second row of A by 1 and subtract it from the third row:

\begin{bmatrix} 1&2&0&2&1 \\ 0&0&1&3&1 \\ 0&0&-3&-9&-3\end{bmatrix}

\Rightarrow \begin{bmatrix} 1&2&0&2&1 \\ 0&0&1&3&1 \\ 0&0&0&0&0\end{bmatrix}

The resulting matrix

U = \begin{bmatrix} 1&2&0&2&1 \\ 0&0&1&3&1 \\ 0&0&0&0&0\end{bmatrix}

is in echelon form with pivots in columns 1 and 3.

Since U has two pivots its rank r = 2. The rank of A is the same as the rank of U so the rank of A is also r = 2. This is the dimension of the column space of A.

The number of columns of A is n = 5 so the dimension of the nullspace of A is n - r = 5 - 2 = 3.

The dimension of the row space of A is the same as that of the column space of A, so the dimension of the row space of A is also r = 2.

Finally, the number of rows of A is m = 3 so the dimension of the left nullspace of A is m - r = 3 - 2 = 1.

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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