Problem 8 (5pts): The transformation T: ?? ? ?? is of the form T(?x) = A?x + ?b where A is an m × n matrix and ?b is a vector in ??. Assuming that ?b is a non-zero vector, show that T is NOT a linear transformation.
Added by Toàn K.
Close
Step 1
We need to show that T is not a linear transformation. **Step 2:** To prove that T is not a linear transformation, we need to show that it does not satisfy the two conditions of linearity: Show more…
Show all steps
Your feedback will help us improve your experience
Madhur L and 88 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ be a basis for a vector space $V,$ and let $T: V \rightarrow W$ be a linear transformation. Prove that if $$T\left(\mathbf{v}_{1}\right)=T\left(\mathbf{v}_{2}\right)=\cdots=T\left(\mathbf{v}_{n}\right)=\mathbf{0}$$ then $T$ is the zero transformation.
General Linear Transformations
a ) Let T : V → W be a function. Prove that T is a linear transformation if and only if for all v_1, """, v_n ∈ V, a_1, """, a_n ∈ ⅂, T (∑_{i=1}^n a_iv_i) = ∑_{i=1}^n a_iT(v_i). (Hint: Use induction!) b.) Let T : ℝ^2 → ℝ^2 be a linear transformation. Show that there exists a 2 × 2 matrix A such that T(v) = Av for all v ∈ ℝ^2.
Vincenzo Z.
Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix. $$A=\left[\begin{array}{r} -3 \\ -2 \\ 0 \\ 1 \end{array}\right]$$.
Linear Transformations
Definition of a Linear Transformation
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD