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 […]

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 […]

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 […]

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 […]