# Recent questions tagged vector-space

1
Let $u$ and $v$ be two vectors in $R^2$ whose Eucledian norms satisfy $\mid u\mid=2\mid v \mid$. What is the value $\alpha$ such that $w=u+\alpha v$ bisects the angle between $u$ and $v$? $2$ $1$ $\dfrac{1}{2}$ $-2$
2
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$ where the polynomial function $f_{a}(x)$ ... $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options
3
Let $V$ be the vector space of all $4 \times 4$ matrices such that the sum of the elements in any row or any column is the same. Then the dimension of $V$ is $8$ $10$ $12$ $14$
4
https://gateoverflow.in/18503/tifr2010-a-11 Here how points are taken and and calculation has been done? Can anybody tell me why (x,y) taking all decimal value? I am not getting , plz somebody explain
5
https://www.youtube.com/watch?v=3SkCNpFOshk In this lecture , can somebody define in 2nd question why $X_{1}+X_{2}\notin V_{1}\cup V_{2}$? I cannot understand the proof
1 vote
6
Let $P$ be a $2$ x $2$ real orthogonal matrix and ${\vec{x}}$ is a real vector $[x_1,x_2]^T$ with length $||{\vec{x}}||$ = ${(x_1^2 + x_2^2)^{1/2}}$. Then which one of the following statements is correct? A. $||P{\vec{x}}||$ $\leq$ $||{\vec{x}}||$ where at ... one vector satisfies $||P{\vec{x}}||$ >$||{\vec{x}}||$ D. No relationship can be established between$||{\vec{x}}||$ and $||P{\vec{x}}||$
1 vote
7
If vectors $\vec{a}=2\hat{i}+\lambda \hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-2\hat{j}+3\hat{k}$ are perpendicular to each other, then value of $\lambda$ is $\dfrac{2}{5}$ $2$ $3$ $\dfrac{5}{2}$
8
Let $u$ and $v$ be two vectors in R2 whose Euclidean norms satisfy $\left \| u \right \| = 2\left \| v \right \|$. What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$? $2$ $\frac{1}{2}$ $1$ $\frac{ -1}{2}$
9
For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, x \rangle}$. Let $a, \: b$ be two vectors in $\mathbb{R} ^n$ ... $a, \: b$? Choose from the following options. ii only i and ii iii only iv only iv and v
1 vote
10
Let $f : X \rightarrow Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if The space $X$ is compact The space $Y$ is compact The space $X$ is complete The space $Y$ is complete
1 vote
11
Let $X$ be a non-empty topological space such that every function $f : X \rightarrow \mathbb{R}$ is continuous. Then $X$ has the discrete topology $X$ has the indiscrete topology $X$ is compact $X$ is not connected
1 vote
12
$X$ is a topological space of infinite cardinality which is homomorphic to $X \times X$. Then $X$ is not connected $X$ is not compact $X$ is not homomorphic to a subset of $R$ None of the above
1 vote
13
$X$ is a metric space. $Y$ is a closed subset of $X$ such that the distance between any two points in $Y$ is at most $1$. Then $Y$ is compact Any continuous function from $Y \rightarrow \mathbb{R}$ is bounded $Y$ is not an open subset of $X$ none of the above
1 vote
14
Let $S$ be a finite subset of $\mathbb{R}^{3}$ such that any three elements in $S$ span a two dimensional subspace. Then $S$ spans a two dimensional space.
1 vote
15
The space of solutions of infinitely differentiable functions satisfying the equation $y" + y = 0$ is infinite dimensional.
1 vote
16
Consider the map $T$ from the vector space of polynomials of degree at most $5$ over the reals to $R \times R$, given by sending a polynomial $P$ to the pair $(P(3), P' (3))$ where $P'$ is the derivative of $P$. Then the dimension of the kernel is $3$.
17
If $V$ is a vector space over the field $\mathbb{Z}/5\mathbb{Z}$ and $\dim_{Z/5\mathbb{Z}}(V)=3$ then $V$ has. 125 elements 15 elements 243 elements None of the above
18
The length of a vector $x = (x_{1},\ldots,x_{n})$ is defined as $\left \| x\right \| = \sqrt{\sum ^{n}_{i=1}x^{2}_{i}}$. Given two vectors $x=(x_{1},\ldots, x_{n})$ and $y=(y_{1},\ldots, y_{n})$, which of the following measures of discrepancy between $x$ and $y$ is ... $\left \| \frac{X}{\left \| X \right \|}-\frac{Y}{\left \| Y \right \|} \right \|$ None of the above.
19
A unit vector perpendicular to both the vectors $a=2i-3j+k$ and $b=i+j-2k$ is: $\frac{1}{\sqrt{3}} (i+j+k)$ $\frac{1}{3} (i+j-k)$ $\frac{1}{3} (i-j-k)$ $\frac{1}{\sqrt{3}} (i+j-k)$
If $V_1$ and $V_2$ are $4$-dimensional subspaces of a $6$-dimensional vector space $V$, then the smallest possible dimension of $V_1 \cap V_2$ is _____.
Consider the set of (column) vectors defined by$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$.Which of the following is TRUE? $\left\{\left[1,-1,0\right]^T,\left[1,0,-1\right]^T\right\}$ is a ... is a linearly independent set, but it does not span $X$ and therefore is not a basis of $X$. $X$ is not a subspace of $R^3$. None of the above
If the linear velocity $\vec V$ is given by $\vec V = x^2y\,\hat i + xyz\,\hat j – yz^2\,\hat k$ The angular velocity $\vec \omega$ at the point $(1, 1, -1)$ is ________