MA25C02: Linear Algebra Question Bank
Unit 1: Vector Spaces | Anna University Regulation 2025
Short Answer Questions
PART A
Vector Spaces
Is it possible for a vector \(u\) in a vector space to have two different negatives? Justify your answer.
Subspaces
Define Subspace of a vector space.
Is the zero vector a subspace? Give reasons for your answer.
Determine whether the set of vectors of the form \((a, b, 1)\) is a subspace of \(\mathbb{R}^3\).
Determine whether the subset \(S = \{ (x, y, 0) \mid x, y \in \mathbb{R} \}\) of the vector space \(V = \mathbb{R}^3\) is a subspace or not.
Is the set of all vectors of the form \((a, 1, 1)\), where \(a\) is real, a subspace of \(\mathbb{R}^3\)? Justify.
Does a line passing through the origin of \(\mathbb{R}^3\) constitute a subspace of \(\mathbb{R}^3\)?
Is the set of all matrices \(A\) such that \(\det(A) = 0\) a subspace of the matrix space \(M_{mn}\)? Justify.
Prove that the intersection of any two subspaces of a vector space \(V\) is also a subspace of \(V\).
If \(V = \mathbb{R}^3\), verify whether \(W = \{ (a_1, a_2, a_3) \mid 2a_1 - 7a_2 + 2a_3 = 0 \}\) is a subspace or not.
Prove that the union of two subspaces of a vector space need not be a subspace.
Linear Combination
For which values of \(k\) will the vector \(v = (1, -2, k)\) in \(\mathbb{R}^3\) be a linear combination of the vectors \(u = (3, 0, -2)\) and \(w = (2, -1, -5)\)?
Consider the vectors \(u = (1, 2, -1)\) and \(v = (6, 4, 2)\). Show that \(w = (9, 2, 7)\) is a linear combination of \(u\) and \(v\).
Linear Dependence & Independence
Determine whether the vectors \(v_1 = (1, -2, 3), v_2 = (5, 6, -1), v_3 = (3, 2, 1)\) are linearly independent or linearly dependent in \(\mathbb{R}^3\).
Why are \(v_1 = (-1, 2, 4)\) and \(v_2 = (5, -10, -20)\) in \(\mathbb{R}^3\) linearly dependent? Explain.
Basis and Dimension
Verify if the vectors \((1, -3, -2), (-3, 1, 3), (-2, 10, -2)\) in \(\mathbb{R}^3\) form a basis for \(\mathbb{R}^3\).
Big Questions & Derivations
PART B
Vector Space Axioms
Determine whether the set of all pairs of real numbers \((x, y)\) with the operations:
\[ (x, y) + (p, q) = (x + p + 1, y + q + 1) \]
\[ k(x, y) = (kx, ky) \]
is a vector space or not. If not, list all the axioms that fail to hold.
Determine whether the set of all pairs of real numbers of the form \((1, x)\) with the operations:
\[ (1, y) + (1, y') = (1, y + y') \]
\[ k(1, y) = (1, ky) \]
is a vector space or not. If not, identify the vector space axioms that fail to hold.
Let \(V\) denote the set of ordered pairs of real numbers. If \((a_1, b_1)\) and \((a_2, b_2)\) are elements of \(V\) and \(c \in \mathbb{R}\), define the operations:
\[ (a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 b_2) \]
\[ c(a_1, b_1) = (ca_1, b_1) \]
Determine whether \(V\) is a vector space over \(\mathbb{R}\) with these operations. If not, list the axioms that fail to hold.
Let \(V\) be the set of all positive real numbers. Define the vector addition and scalar multiplication as follows:
\[ x + y = xy \]
\[ \alpha x = x^\alpha \]
Determine whether or not \(V\) is a vector space over \(\mathbb{R}\) with respect to the above operations, where \(\alpha\) is a real number.
Check whether the set of all pairs of real numbers of the form \((x, y)\) with the operations:
\[ (x, y) + (x', y') = (xx', yy') \]
\[ k(x, y) = (kx, ky) \]
is a vector space.
Matrix Spaces
Show that the set \(V\) of all \(2 \times 2\) matrices with real entries is a vector space if addition is defined to be matrix addition and scalar multiplication is defined to be matrix scalar multiplication.
Determine whether the set of all \(2 \times 2\) matrices of the form \(\begin{bmatrix} a & 1 \\ 1 & b \end{bmatrix}\), where \(a\) and \(b\) are real, with standard matrix addition and scalar multiplication is a vector space or not. If not, list all axioms that fail to hold.
Determine whether the set of all \(2\times 2\) matrices of the form \(\begin{bmatrix} a & a+b \\ a+b & b \end{bmatrix}, a,b \in \mathbb{R}\) with respect to standard matrix addition and scalar multiplication is a vector space or not. If not, list all the axioms that fail to hold.
Subspaces
Let \(V = \mathbb{R}^3\), \(W_1 = \{(x,x,x) \mid x \in \mathbb{R}\}\) and \(W_2 = \{(0,y,z) \mid y, z \in \mathbb{R}\}\) be two subspaces of \(V\). Prove that \(V = W_1 \oplus W_2\).
Linear Combination & System of Linear Equations
Let \(a_1 = \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix}\), \(a_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}\), \(a_3 = \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix}\), and \(b = \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}\) be vectors. Check whether \(b\) is a linear combination of \(a_1, a_2\) and \(a_3\) or not.
Verify whether the first polynomial can be expressed as a linear combination of the other two in \(P_3(\mathbb{R})\) for the polynomials \(x^3-8x^2+4x\), \(x^3-2x^2+3x-1\) and \(x^3-2x+3\).
Linear Span
Let \(v_1 = (2,1,0,3), v_2 = (3, -1, 5, 2)\) and \(v_3 = (-1,0,2,1)\). Does \((2,3, -7, 3)\) belong to the span of \(\{v_1,v_2,v_3\}\)? Justify your answer.
Linear Independence & Dependence
Verify whether the set \(S = \left\{ \begin{pmatrix} 1 & -3 & 2 \\ -4 & 0 & 5 \end{pmatrix}, \begin{pmatrix} -3 & 7 & 4 \\ 6 & -2 & -7 \end{pmatrix}, \begin{pmatrix} -2 & 3 & 11 \\ -1 & -3 & 2 \end{pmatrix} \right\}\) in \(M_{2\times 3}(\mathbb{R})\) is linearly dependent or not.
Show that the three vectors \(u=(0,3,1,-1), v=(6,0,5,1)\) and \(w=(4,-7,1,3)\) form a linearly dependent set in \(\mathbb{R}^4\).
Determine if the given set in \(P_4(\mathbb{R})\) is linearly independent or linearly dependent: \(x^4-x^3+5x^2-8x+6\), \(-x^4+x^3-5x^2+5x-3\), \(x^4+3x^2-3x+5\) and \(2x^4+x^3+4x^2+8x\).
Basis & Dimension
Find a basis and the dimension of the solution space \(W\) of the system:
\[
\begin{aligned}
2x_1 + 2x_2 - x_3 + x_5 &= 0 \\
-x_1 - x_2 + 2x_3 - 3x_4 + x_5 &= 0 \\
x_1 + x_2 - 2x_3 - x_5 &= 0 \\
x_3 + x_4 + x_5 &= 0
\end{aligned}
\]
Let \(v_1= (1,2,1), v_2= (2,9,0)\) and \(v_3 = (3,3,4)\). Show that the set \(S = \{ v_1,v_2,v_3 \} \) is a basis for \(\mathbb{R}^3\).
Obtain the solution space, basis, and dimension for the system:
\(\begin{bmatrix} 1 & -3 & 4 & -2 & 5 & 4 \\ 2 & -6 & 9 & -1 & 8 & 2 \\ 2 & -6 & 9 & -1 & 9 & 7 \\ -1 & 3 & -4 & 2 &-5 & -4 \end{bmatrix}\).
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