How To Solve Vector Problems

(c) If $\mathbf_1, \mathbf_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf_1, A\mathbf_2$ are also linearly independent.Read solution (a) For what value(s) of $a$ is the following set $S$ linearly dependent?

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See for example https://en.wikipedia.org/wiki/Absolute_bearing#/media/File: Compass_Card_B+That means that you can decompose the velocity of the wind into two components, one towards North, with magnitude $23\cos$mph, and one towards East, with magnitude $23\sin$mph.

You need to add these velocities to the velocity of the plane.

For each of the following vectors, determine whether the vector is in the nullspace $\cal N(A)$.

(a) $\begin -3 \\ 0 \\ 1 \\ 0 \end$ (b) $\begin -4 \\ -1 \\ 2 \\ 1 \end$ (c) $\begin 0 \\ 0 \\ 0 \\ 0 \end$ (d) $\begin 0 \\ 0 \\ 0 \end$ Then, describe the nullspace $\cal N(A)$ of the matrix $A$.

Whats the correct answer and correct way of doing this?

Bearing in this case means that the wind blows almost towards East (slightly on the northern side of it).Solve the following system of linear equations and give the vector form for the general solution.\begin x_1 -x_3 -2x_5&=1 \ x_2 3x_3-x_5 &=2 \ 2x_1 -2x_3 x_4 -3x_5 &= 0 \end (The Ohio State University, linear algebra midterm exam problem) Read solution Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let \[W=\] be a subset of $V$.Find the groundspeed and resulting bearing of the plane.I tried doing it this way as I was not given an angle for the airspeed of the plane for the first one, I don't know if the component form of the wind is correct or not, I'm confused on the idea of converting the bearing of an angle.Read solution In this problem, we use the following vectors in $\R^2$.\[\mathbf=\begin 1 \ 0 \end, \mathbf=\begin 1 \ 1 \end, \mathbf=\begin 2 \ 3 \end, \mathbf=\begin 3 \ 2 \end, \mathbf=\begin 0 \ 0 \end, \mathbf=\begin 5 \ 6 \end.\] For each set $S$, determine whether $\Span(S)=\R^2$.Since the plane is moving South with respect to the air, the new components along ground are going to be 2-23\cos$mph towards South and \sin$mph towards East.Let $A=\begin 1 & 0 & 3 & -2 \ 0 &3 & 1 & 1 \ 1 & 3 & 4 & -1 \end$.Read solution For each of the following matrix $A$, prove that $\mathbf^A\mathbf \geq 0$ for all vectors $\mathbf$ in $\R^2$. Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix.Also, determine those vectors $\mathbf\in \R^2$ such that $\mathbf^A\mathbf=0$. Read solution Problem 1 Let $W$ be the subset of the $-dimensional vector space $\R^3$ defined by \[W=\left\.\] (a) Which of the following vectors are in the subset $W$? \[(1) \begin 0 \ 0 \ 0 \end \qquad(2) \begin 1 \ 2 \ 2 \end \qquad(3)\begin 3 \ 0 \ 0 \end \qquad(4) \begin 0 \ 0 \end \qquad(5) \begin 1 & 2 & 4 \ 1 &2 &4 \end \qquad(6) \begin 1 \ -1 \ -2 \end.\] (b) Determine whether $W$ is a subspace of $\R^3$ or not.