Marenonigt

2022-01-06

If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it.

David Clayton

Beginner2022-01-07Added 36 answers

By definition-

If V be a vector space over an arbitray field F, then we say that V is finite dimensional if it is spanmed by a finite set of vectors.

Let, dimV =n

say

Now, as W is a stubspace of then W is spaned by at most n elements of the set S.

Hence, by definition of finite dimensional vector sapace- W is finite dimensional.

An object moving in the xy-plane is acted on by a conservative force described by the potential energy function

where$U(x,y)=\alpha (\frac{1}{{x}^{2}}+\frac{1}{{y}^{2}})$ is a positive constant. Derivative an expression for the force expressed terms of the unit vectors$\alpha$ and$\overrightarrow{i}$ .$\overrightarrow{j}$ I need to find a unique description of Nul A, namely by listing the vectors that measure the null space

?

$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$T must be a linear transformation, we assume. Can u find the T standard matrix.$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$

?Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR

Consider the points below

P(1,0,1) , Q(-2,1,4) , R(7,2,7).

a) Find a nonzero vector orthogonal to the plane through the points P,Q and R.

b) Find the area of the triangle PQR.Consider two vectors A=3i - 1j and B = - i - 5j, how do you calculate A - B?

Let vectors A=(1,0,-3) ,B=(-2,5,1) and C=(3,1,1), how do you calculate 2A-3(B-C)?

What is the projection of $<6,5,3>$ onto $<2,-1,8>$?

What is the dot product of $<1,-4,5>$ and $<-5,7,3>$?

Which of the following is not a vector quantity?

A)Weight;

B)Nuclear spin;

C)Momentum;

D)Potential energyHow to find all unit vectors normal to the plane which contains the points $(0,1,1),(1,-1,0)$, and $(1,0,2)$?

What is a rank $1$ matrix?

How to find unit vector perpendicular to plane: 6x-2y+3z+8=0?

Can we say that a zero matrix is invertible?

How do I find the sum of three vectors?

How do I find the vertical component of a vector?