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 dependent and Linear Indedependent
✓
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.
Sub Spaces
✓
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 and 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 and Linear 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:
\(x_1+2x_2+2x_3-x_4 + 3x_5 = 0\); \(x_1+2x_2+3x_3 + x_4 + x_5 = 0\); \(3x_1+6x_2+8x_3 + x_4 + 5x_5 = 0\).
✓
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:
\(\begin{bmatrix} 1 & -3 & 4 & -2 & 5 & 4 \\ 2 & -6 & 9 & -1 & 8 & 2 \\ 2 & -6 & 9 & -1 & 9 & 7 \\ -1 & 3 & -4 & 2 &-5 & -4 \end{bmatrix}\).
Looking for PDF solutions? Engineering Mathematica is updated daily for Reg 2025!
No comments:
Post a Comment