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Which of the following transformations are linear?

Verify which of the following transformations are linear. $T_1(x_1,x_2,x_3) = (x_1,0,x_3)$ $T_2(x_1,x_2)=(2x_1 – 3x_2,x_1 +4,5x_2)$ $T_3(x_1,x_2,x_3)=(1,x_2,x_3)$ $T_4(x_1,x_2)=(4x_1 – 2x_2,3|x_2|)$ $T_5(x_1,x_2,x_3)=(x_1,x_2,-x_3)$ The objective of this question is to find the linear transformation from the given transformation. This question uses the concept of linear transformation. The linear transformation is the mapping of one vector space to another […]

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 scalar and vector projections of b onto a. a=i+j+k, b=i−j+k

The aim of this question is to find the Scalar and Vector Projection of the given two vectors. The basic concept behind this article is the understanding of Scalar and Vector Projections of vector quantities and how to calculate them. The Scalar Projection of one vector $vec{a}$ onto another vector $vec{b}$ is expressed as the […]