Question

3. Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation, and suppose that $v_1 \in \mathbb{R}^n$ is a linear combination of $v_2$ and $v_3$. Show that $T(v_1)$ is also a linear combination of $T(v_2)$ and $T(v_3)$. [3 marks] $v_1 = x_2\vec{v_2} + x_3\vec{v_3}$ $T(\vec{v_1}) = T(\vec{v_2}) + T(\vec{v_3})$ according to a linear transformation Property $T(\vec{e_1}) = T(\vec{e_2}) + T(\vec{e_3})$

          3. Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation, and suppose that $v_1 \in \mathbb{R}^n$ is a linear combination of $v_2$ and $v_3$. Show that $T(v_1)$ is also a linear combination of $T(v_2)$ and $T(v_3)$. [3 marks]
$v_1 = x_2\vec{v_2} + x_3\vec{v_3}$
$T(\vec{v_1}) = T(\vec{v_2}) + T(\vec{v_3})$
according to a linear transformation
Property
$T(\vec{e_1}) = T(\vec{e_2}) + T(\vec{e_3})$

3. Let T: ℝ^n →ℝ^m be a linear transformation, and suppose that v1 ∈ℝ^n is a linear combination of v2 and v3. Show that T(v1) is also a linear combination of T(v2) and T(v3). [3 marks]
v1 = x2v⃗2⃗ + x3v⃗3⃗
T(v⃗1⃗) = T(v⃗2⃗) + T(v⃗3⃗)
according to a linear transformation
Property
T(e⃗1⃗) = T(e⃗2⃗) + T(e⃗3⃗)

Added by Raymond J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/28/2023

Step 1

Step 1: Given that v1 is a linear combination of v2 and v3, we can write: v1 = x2v2 + x3v3 where x2 and x3 are scalars. Show more…

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3. Let T: R" → Rm be a linear transformation, and suppose that v1∈ R" is a linear combination of v2 and v3. Show that T(v1) is also a linear combination of T(v2) and [3 marks] V₁ = x2 + x33 T(Vi) = T(V2)+T(V3) according to a linear transformation Property T(e') = T(e²) + T (e3) 2