Linear Algebra Midterm Practice Questions

204 questions are ready for this course, written from real past papers and checked by a second, independent model before they reach you. Below are the topics that keep coming up, and five questions you can try right now.

What the exam asks

Matrix Inverse15
Linear Independence And Basis14
Linear Independence And Span14
Determinant Calculation13
Solving Linear System12
Transition Matrix12
Linear Independence10
Rref And Homogeneous System10
Determinant And Inverse9
Null And Column Space8
Change Of Basis7
System Consistency7

…and 73 more across smaller topics.

Five questions from the pool

1Linear Independence And Span · medium

Let

S = \left\{ \begin{bmatrix} 1 \\ 2 \\ 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \\ 6 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 5 \\ 1 \end{bmatrix} \right\}

(a) Determine if the set $S$ is linearly independent.
(b) Determine if

v = \begin{bmatrix} 1 \\ 0 \\ 4 \\ 3 \end{bmatrix}

is in $\text{Span}(S)$ .

3Solving Systems Via Rref · medium

Consider the following homogeneous system of linear equations:

\begin{cases} 2x_1 - x_2 + 3x_3 - x_4 + 2x_5 = 0 \\ 4x_1 - 2x_2 + 6x_3 - 2x_4 + 4x_5 = 0 \\ x_1 - 2x_2 + 5x_3 + 4x_4 - x_5 = 0 \end{cases}

(a) Write the coefficient matrix of the system.
(b) Find the Reduced Row Echelon Form (RREF) of the coefficient matrix.
(c) Use the RREF to find the general solution to the system.

5Transition Matrix · medium

Let $V = \mathbb{P}_1$ and $\mathcal{C} = \{3+2x, 1-x\}$ , $\mathcal{D} = \{2, 4+x\}$ be two ordered bases.
(a) Find the transition matrix from $\mathcal{D}$ to $\mathcal{C}$ , i.e. $[I]_{\mathcal{D}}^{\mathcal{C}}$ .
(b) Let $q(x) = 2x - 1$ . Find $[q(x)]_{\mathcal{D}}$ and use the transition matrix to find $[q(x)]_{\mathcal{C}}$ .

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