Let T : R^4 -> R^3 be the linear transformation defined by T(w, x, y, z) = (w + x, y + z, w + y). a. Find a basis for ker T. What is dim(ker T)? b. Find a basis for im T. What is dim(im T)? c. Is T one-to-one? d. Is T onto?
Added by Cynthia F.
Close
Step 1
Step 1:** To find a basis for ker T, we need to solve the system of equations: \[ \begin{cases} w + x = 0 \\ y + z = 0 \\ w + y = 0 \end{cases} \] ** Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 67 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
Find basis for ken(T) and range(T)
Nick J.
Show that the function T Mzz is a linear transformation: Given by T T(z) = (c - 2d)z^2 + (4d - 2c)z + 3b + a For the linear transformation stated in problem 2 above: Find the basis of ker(T) What is the nullity(T)? Is T one-to-one? Why or why not?
Madhur L.
In each case, (i) find a basis of ker T, and (ii) find a basis of im T. You may assume that T is linear. a. T : P2 → R2; T(a + bx + cx2) = (a, b) b. T : P2 → R2; T(p(x)) = (p(0), p(1)) c. T : R3 → R3; T(x, y, z) = (x + y, x + y, 0) d. T : R3 → R4; T(x, y, z) = (x, x, y, y) e. T : M22 → M22; T [a b; c d] = [a + b b + c; c + d d + a] f. T : M22 → R; T [a b; c d] = a + d g. T : Pn → R; T(r0 + r1x + ⋯ + rnxn) = rn h. T : Rn → R; T(r1, r2, …, rn) = r1 + r2 + … + rn
Sri K.
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