Linear Algebra Important Questions & Notes – Regulation 2025 Engineering Mathematics

Anna University – Linear Algebra Important Questions (Regulation 2025) 

UNIT I-VECTOR SPACE

PART A – 2 Marks Questions

Vector Spaces

1. Is it possible for a vector $u$ in a vector space to have two different negatives? Justify.

Sub Spaces

2. Is the zero vector a subspace? Give reason.
3. Determine whether the set of vectors of the form $(a,b,1)$ is a subspace of $R^3$.
4. Determine whether the subset $S=\{(x,y,0)\}$ of $V=R^3$ is a subspace.
5. Is the set $(a,1,1)$ a subspace of $R^3$? Justify.
6. Does a line passing through origin in $\mathbb{R}^3$ form a subspace?
7. Is the set of matrices $A$ such that $\det(A)=0$ a subspace of $M_{mn}$?
8. Prove that intersection of two subspaces of $V$ is a subspace of $V$.

Linear Combination & System of Linear Equations

9. For which values of $k$ will $v=(1,-2,k)$ be a linear combination of $u=(3,0,-2)$ and $w=(2,-1,-5)$?
10. Show that $w=(9,2,7)$ is a linear combination of $u=(1,2,-1)$ and $v=(6,4,2)$.
11. Can $(-2,0,3)$ be written as a linear combination of $(1,3,0)$ and $(2,4,-1)$? Justify.

Linear Independence & Dependence

12. Determine whether $(1,-2,3)$, $(5,6,-1)$ and $(3,2,1)$ are linearly independent in $R^3$.
13. Why are $(-1,2,4)$ and $(5,-10,-20)$ linearly dependent?
14. Do $(1,1,2)$, $(1,0,1)$ and $(2,1,3)$ span $R^3$?

PART B – Marks Questions

Vector Spaces

1. Check whether $(x,y)+(p,q)=(x+p+1,y+q+1)$ and $k(x,y)=(kx,ky)$ define a vector space.
2. Check whether $(1,x)$ with given operations forms a vector space.
3. Show that set of all $2 \times 2$ matrices forms a vector space.
4. Is $\begin{bmatrix} a & 1 \\ 1 & b \end{bmatrix}$ a vector space under standard operations?
5. Determine whether $V$ with custom operations forms a vector space over $\mathbb{R}$.

Sub Spaces

6. Let $V=R^3$, $W_1=\{(x,x,x)\}$ and $W_2=\{(0,y,z)\}$. Prove that $V=W_1 \oplus W_2$.

Linear Combination

7. Show that $(9,2,7)$ is a linear combination of $(1,2,-1)$ and $(6,4,2)$.
8. Check whether $b$ is a linear combination of $a_1,a_2,a_3$.

Linear Independence

9. Verify whether given matrices in $M_{2\times3}(R)$ are linearly dependent.
10. Show that $(0,3,1,-1)$, $(6,0,5,1)$ and $(4,-7,1,3)$ are linearly dependent in $\mathbb{R}^4$.
11. Check whether $(2,3,-7,3)$ is in span of $\{v_1,v_2,v_3\}$.

Basis and Dimension

12. Find basis and dimension of given homogeneous system.
13. Find basis of solution space of homogeneous equations.
14. Find basis for space spanned by given vectors.
15. Show that $(1,2,1),(2,9,0),(3,3,4)$ form a basis for $R^3$.
16. Find dimension and basis of solution space $W$.
17. Obtain solution space and its dimension from given matrix.
18. Show that $S_1$ is a minimal generating set.

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👉 Internal Links:

• Vector Space Definition & Axioms
• Linear Independence Explained
• Basis and Dimension Problems