Problems 4 I-48. compute the image under T of u, and find the vector(s), if any, that are mapped

step 1 So we're given these two figures and we want to try to plot T of W. So the hint tells us to firs

Problems 4 I-48. compute the image under T of u, and find...

Problems $4 I-48$. compute the image under $T$ of $\overrightarrow{\mathrm{u}}$, and find the vector(s), if any, that are mapped to $\overrightarrow{\mathbf{w}}$. $T\left(u_1, u_2, u_3\right)=\left(u_1+u_3, u_2-u_3\right)$, $$ \overrightarrow{\mathbf{u}}=(1,1,1), \overrightarrow{\mathbf{w}}=(0,0) $$

Problems $4 I-48$. compute the image under $T$ of $\overrightarrow{\mathrm{u}}$, and find the vector(s), if any, that are mapped to $\overrightarrow{\mathbf{w}}$.
$T\left(u_1, u_2, u_3\right)=\left(u_1+u_3, u_2-u_3\right)$,
$$
\overrightarrow{\mathbf{u}}=(1,1,1), \overrightarrow{\mathbf{w}}=(0,0)
$$

Key Concepts

Kernel (Null Space) of a Linear Transformation

The kernel, or null space, of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain. Finding the kernel involves solving the equation T(x) = 0, where 0 is the zero vector. This concept is particularly important in determining whether a transformation is injective (one-to-one) and in understanding the structure of the transformation's solutions.

Image of a Vector

The image of a vector under a linear transformation is the result obtained when the transformation is applied to that vector. It involves substituting the components of the vector into the transformation's formula to yield another vector in the target vector space. Understanding how to compute the image of a vector helps in analyzing the behavior of the transformation, such as determining the outputs corresponding to given inputs.

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, if T is a linear transformation, then for any vectors v and w and any scalar c, T(v + w) = T(v) + T(w) and T(cv) = cT(v). This concept is fundamental in linear algebra as it allows one to understand how vectors are mapped from one space to another while maintaining their linear structure.

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