### Find an explicit description of nul A by listing vectors that span the null space.

begin{equation*} A = begin{bmatrix} 1 & 2 & 3 & -7 \ 0 & 1 & 4 & -6 end{bmatrix} end{equation*} This problem aims to find the vectors in matrix A that span the null space. Null space of matrix A can be defined as the set of n column vectors x such that their […]

### Find an orthogonal basis for the column space of the matrix given below:

[ boldsymbol{ left[ begin{array}{ccc} 3 & -5 & 1 \ 1 & 1 & 1 \ -1 & 5 & -2 \ 3 & -7 & -8 end{array} right] }]This question aims to learn the Gram-Schmidt orthogonalization process. The solution given below follows the step-by-step procedure. In Gram-Schmidt orthogonalization, we assume the first basis vector […]

### Find the rate of change of f at p in the direction of the vector u.

[f(x,y,z) = y^2e^{xyz}, P(0,1,-1), u = <frac{3}{13},frac{4}{13},frac{12}{13}>] This question aims to find the rate of change or gradient and projections of vector spaces onto a given vector. Gradient of a vector can be found using following formula: [nabla f(x,y,z) = bigg ( frac{partial f}{partial x} (x,y,z),frac{partial f}{partial y} (x,y,z),frac{partial f}{partial z} (x,y,z) bigg )] Projection of a […]

### Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers

For the given set of all vectors shown as $W=left[ begin{matrix}4a + 3b\0\ begin{matrix}a+b+c\c – 2a\end{matrix}\end{matrix}right]$, and here a, b and c are arbitrary real numbers. Find vector set S which spans W or give an example to show that W is not a space vector. In this question, we have to find […]

### Find the values of x such that the angle between the vectors (2, 1, -1) and (1, x, 0) is 40.

The question aims to find the value of an unknown variable given in 3D vector coordinates and the angle between those vectors. The question depends on the dot product of two 3D vectors to calculate the angle between those vectors. As the angle is already given, we can use the equation to calculate the unknown […]

### Find a single vector x whose image under t is b

Transformation is defined as T(x)=Ax, find whether x is unique or not. [A=begin{bmatrix} 1 & -5 & -7\ 3 & 7 & 5end{bmatrix}] [B=begin{bmatrix} 2\ 2end{bmatrix}] This question aims to find the uniqueness of vector $x$ with the help of linear transformation. This question uses the concept of Linear transformation with reduced row echelon form. Reduced row echelon form […]

### Find the change of coordinates matrix from B to the standard basis in R^n.

[ boldsymbol{ B = left{ Bigg [ begin{array}{c} 1 \ -2 \ 5 end{array} Bigg ] , Bigg [ begin{array}{c} 3 \ 0 \ -1 end{array} Bigg ] , Bigg [ begin{array}{c} 8 \ -2 \ 7 end{array} Bigg ] right} } ] The aim of this question is to find the change-of-coordinates matrix given […]

### Compute the distance d from y to the line through u and the origin.

[ y = begin {bmatrix} 5 \ 3 end {bmatrix} ] [ u = begin {bmatrix} 4 \ 9 end {bmatrix} ] The question aims to find the distance between vector y to the line through u and the origin. The question is based on the concept of vector multiplication, dot product, and orthogonal projection. […]

### Find the vectors T, N, and B, at the given point.

[ R(t) = < t^{2}, frac{2}{3} t^{3} , t > text {and point} < 4, frac{-16}{3}, -2 > ] This question aims to determine the tangent vector, normal vector, and the binormal vector of any given vector. The tangent vector T is a vector that is tangent to the given surface or vector at any […]

### Find the dimension of the subspace spanned by the given vectors:

[ begin{bmatrix} 2 \ 4 \ 0 end{bmatrix} , begin{bmatrix} -1\ 6 \ 2 end{bmatrix} , begin{bmatrix} 1 \ 5 \ -3 end{bmatrix} , begin{bmatrix} 7 \ 2 \ 3 end{bmatrix} ] The question aims to find the dimension of the subspace spanned by the given column vectors. The background concepts needed for this question […]

### Which of the following expressions are meaningful which are meaningless explain:

(a . b) . c (a . b) c |a|(b . c) a . (b + c) a . b + c |a| . (b+c) The questions aim to find the expressions of some vector multiplication and addition to check whether the expression is meaningful or meaningless. The background concept needed for this question to […]

### Sketch the vector field f by drawing a diagram like this figure. f(x, y) = (yi + xj)/(x2 + y2)

The aim of this question is to develop understanding by visualizing the flow of vector fields. To draw a vector field, we use the following steps: a) Convert the given function in the vector notation (vector components form). b) Define some arbitrary points in the vector space. c) Evaluate vector values at each of these […]

### For the two vectors in the figure (Figure 1) , find the magnitude of the vector product

– $overrightarrow A space times overrightarrow B$ – Determine the vector product’s direction  $overrightarrow A space times overrightarrow B$. – Calculate the scalar product when the angle is $60 { circ}$ and the vector magnitude is $5 and 4$. – Calculate the scalar product when the angle is […]

### Find the best approximation to z by vectors of the form c1v1 + c2v2

This problem aims to find the best approximation to a vector $z$ by a given combination of vectors as $c_1v_1 + c_2v_2$, which is same as the vectors $v_1$ and $v_2$ in span. For this problem, you should know about the best approximation theory, fixed point approximation, and orthogonal projections. We can define fixed-point theory as […]

### Let f be a fixed 3×2 matrix, and H be the set of matrices A belonging to a 2×4 matrix. If we assume that the property FA = O holds true, show that H is a subspace of M2×4. Here O represents a zero matrix of order 3×4.

The aim of this question is to comprehend the key linear algebra concepts of vector spaces and vector subspaces. A vector space is defined as a set of all vectors that fulfill the associative and commutative properties for vector addition and scalar multiplication operations. The minimum no. of unique vectors required to describe a certain vector space is called basis vectors. A vector space is an n-dimensional space defined by linear combinations of basis vectors. Mathematically, a […]

### Find a basis for the eigenspace corresponding to each listed eigenvalue of A given below:

[ boldsymbol{ A = left[ begin{array}{cc} 1 & 0 \ -1 & 2 end{array} right], lambda = 2, 1 } ] The aim of this question is to find the basis vectors that form the eigenspace of given eigenvalues against a specific matrix. To find the basis vector, one only needs to solve the following system for $x$: [ A […]

### Find, correct to the nearest degree, the three angles of the triangle with the given vertices. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3).

The main objective of this question is to find the three angles of a triangle given three vertices. The angles can be found using the dot product of the vectors representing the sides of the triangle. A triangle is a polygon with three-sides that is also referred to as a trigon. Every triangle possesses $3$ sides and […]

### Find the vectors T, N, and B at the given point. r(t)=< t^2,2/3 t^3,t > and point < 4,-16/3,-2 >.

This question aims to find the Tangent, Normal, and Binormal vectors by using the given point and a function. Consider a vector function, $vec{r}(t)$. If $vec{r}'(t)neq 0$ and $vec{r}'(t)$ exist then $vec{r}'(t)$ is called a tangent vector. The line that passes through the point $P$ and is parallel to the tangent vector, $vec{r}'(t)$, is the […]

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

Take note of the following points: $P(1,0,1) , Q(-2,1,4) , R(7,2,7)$ Find a nonzero vector orthogonal to the plane through the points $P, Q$, and $R$. Find the area of the triangle $PQR$. The purpose of this question is to find an orthogonal vector and the area of a triangle using the vectors $P, Q,$ […]