List of top Mathematics Questions on Linear Algebra asked in CPET

If \(\lambda_1,\lambda_2,\lambda_3\) are the eigenvalues of the matrix \[\begin{pmatrix}-2&2&-3\\2&1&-6\\-1&-2&0\end{pmatrix}\] then the value of \(\lambda_1^2+\lambda_2^2+\lambda_3^2\) is ____.

If the eigenvalue \(\lambda\) of a matrix \(M\) has algebraic multiplicity 3 and geometric multiplicity 2, then which one of the following options is true?

Which one of the following matrices has \(p(x) = x^3 - 8x^2 + 5x + 7\) as the minimal polynomial?

Let \((V, \langle,\rangle)\) be an inner product space and let \(\langle x,y\rangle = \|x\|\|y\|\) for all \(x,y \in V\). Then which one of the following options is true?

The projection of the vector \(u=(2,-1,3) \in \mathbb{R}^3\) onto the vector \(v=(1,2,-1)\) of the vector space \(\mathbb{R}^3\) is ____.

Which one of the following statements is false?
(I) The linear transformation \(L:\mathbb{R}^2 \to \mathbb{R}^2\) that reflects a vector across the line \(y=x\) is both self-adjoint and normal.
(II) The linear transformation \(L:\mathbb{R}^2 \to \mathbb{R}^2\) which rotates a vector by an angle \(\pi/4\) in the anti-clockwise direction is both self-adjoint and normal.

Let V be a 7-dimensional vector space. If U, W are subspaces of V with dimensions 4 and 5 respectively, then which of the following is NOT a possible value of dimension of \(U \cap W\)?

For a linear transformation \(L:V\to W\) with \(\dim(V)=10\) and \(\dim(W)=8\), which one of the following options is correct?

Let \(L\) be a linear transformation defined on the vector space \(\mathbb{R}^6\) over the field \(\mathbb{R}\). If \(\operatorname{Rank}(L)=4\) and \(\operatorname{Nullity}(L^2)=3\), then \(\operatorname{Rank}(L^2)\) is ____ (\(L^2\) denotes the composition of \(L\) with itself).

On the vector space \(\mathbb{R}^3\) over the field \(\mathbb{R}\) with the standard basis \(\{e_1,e_2,e_3\}\), what is the annihilator \((S^0)\) of the subspace \(S=\text{span}\{e_1,e_2\}\)?

Let V be the real vector space consisting of all polynomials in one real variable with real coefficients of degree at most 6, including the zero polynomial. Then which one of the following options is true?

For a real \(4 \times 3\) matrix \(M\) and the standard basis \(\{e_1,e_2,e_3\}\) of \(\mathbb{R}^3\), which of the following statements is/are true?
(I) If \(\text{rank}(M)=1\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).
(II) If \(\text{rank}(M)=2\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).
(III) If \(\text{rank}(M)=3\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).