### 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 […]

### Find a vector $A$ with representation given by the directed line segment $AB$. Draw $AB$ and the equivalent representation starting from the origin $A(4, 0, -2), B(4, 2 ,1)$.

The aim of this question is to become familiar with the vector representation. Two vectors are given in this question and their product needs to be found. After that, the visual representation of the origin is also made. This question is based on the concepts of physics. Vectors are quantities that have magnitude as well […]

### 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 […]