Linear Independence and Rank

Practice Questions

Linear Algebra › Linear Independence and Rank

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1

Find the rank of the following matrix.

$\begin{bmatrix} 10 & 2 \ 4 & 5\ 5& 10 \end{bmatrix}$

2

True or False: If a $5 \times 5$ matrix has linearly independent columns, then .

3

Consider the following set of vectors

$\b{v}_1= (1,0,0)$

$\b{v}_2= (0,1,0)$

$\b{v}_3= (0,0,1)$

Is the the set linearly independent?

4

The rank and the nullity of a matrix with four rows and six columns are the same. What number do they share?

5

Calculate the Rank of the following matrix

$\begin{bmatrix} 1&2 &3 \ 1& 2& 3\ 1&2 &3 \end{bmatrix}$

6

The nullity of a matrix is 3; the nullity of $A^{T }$ is also 3.

True, false or undetermined: is a square matrix.

7

Consider the following set of vectors

$\b{v}_1= (1,0,0)$

$\b{v}_2= (0,1,0)$

$\b{v}_3= (0,0,1)$

$\v_4 = (2, 1, 5)$ $\b{v}_4 = (2,1,5)$

Is the the set linearly independent?

8

The rank and nullity of a matrix are 4 and 2, respectively. The nullity of $A^{T }$ is 3. What are the dimensions of ?

9

Determine if the following matrix is linearly independent or not.

$A=\begin{bmatrix} 5 & 3\ 10 & 6 \end{bmatrix}$

10

$A = \begin{bmatrix} 1 & 0 & 2 \4 & 3 & 11 \ 2 & 2 & 6 \-3 & 1 & -5 \ 3 & -2 & 4\end{bmatrix}$

What are the rank and nullity of ?

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