List of top Mathematics Questions on Vector space asked in CUET (PG)

Match the LIST-I with LIST-II - For a linear transformation $T : U(F) \to V(F)$

Choose the correct answer from the options given below:

Let $T : U(F) \to V(F)$ be a linear transformation, then which of the following holds:
A. $\text{Rank}(T) + \text{dim}(V) = \text{Nullity}(T)$ B. $\text{Rank}(T) + \text{Nullity}(T) = \text{dim}(U)$ C. $\text{Nullity}(T) = \text{dim}(V)$ D. $\text{Nullity}(T) = \text{dim}(\text{Ker}(T))$ E. $\text{Nullity}(T) = \text{dim}(U)$ Choose the correct answer from the options given below:

Which of the following functions $T : \mathbb{R}^2 \to \mathbb{R}^2$ are linear transformations? A. $T(x_1, x_2) = (x_1, x_1 + x_2)$
B. $T(x_1, x_2) = (x_1 + x_2, x_2)$
C. $T(x_1, x_2) = (x_1 + 1, x_2 + 2)$
D. $T(x_1, x_2) = (|x_1| - |x_2|, 0)$
E. $T(x_1, x_2) = (x_1 - x_2, x_1 + x_2)$
Choose the correct answer from the options given below:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Set $\{(1,0,0), (0,1,0), (0,0,1)\}$ forms a basis of $V_3(\mathbb{R})$. Reason R : Let $V$ be a vector space over a field $F$ and $S$ be a non-empty subset of $V$, then $S$ is a basis of $V$ if
(i) $S$ is linearly dependent
(ii) $\text{span}(S) = V$. In the light of the above statements, choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $P_0(t) = 1, P_1(t) = t, P_2(t) = t^2, P_3(t) = t^3$, then $\{P_0, P_1, P_2, P_3\}$ is linearly independent. Reason R : $C_0 P_0 + C_1 P_1 + C_2 P_2 + C_3 P_3 = 0 \implies C_0 = 0, C_1 = 0, C_2 = 0, C_3 = 0$.
In the light of the above statements, choose the correct answer from the options given below

If $W_1$ and $W_2$ are finite dimensional subspaces of a vector space $V$, then: