Finding the Image Two Ways In Exercises 57 and 58, find T ( v ) by using (a) the standard and (b) the matrix relative to B and B ′ . T : R 3 → R 3 T ( x , y ) = ( − x , y , x + y ) , v = ( 0 , 1 ) B = { ( 1 , 1 ) , ( 1 , − 1 ) } , B ′ = { ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 1 , 0 , 0 ) }
Finding the Image Two Ways In Exercises 57 and 58, find T ( v ) by using (a) the standard and (b) the matrix relative to B and B ′ . T : R 3 → R 3 T ( x , y ) = ( − x , y , x + y ) , v = ( 0 , 1 ) B = { ( 1 , 1 ) , ( 1 , − 1 ) } , B ′ = { ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 1 , 0 , 0 ) }
Solution Summary: The author explains how to find the matrix T(v) by using the standard matrix for the given condition.
Q1: A: Let M and N be two subspace of finite dimension linear space X, show that if M = N
then dim M = dim N but the converse need not to be true.
B: Let A and B two balanced subsets of a linear space X, show that whether An B and
AUB are balanced sets or nor.
Q2: Answer only two
A:Let M be a subset of a linear space X, show that M is a hyperplane of X iff there exists
ƒ€ X'/{0} and a € F such that M = (x = x/f&x) = x}.
fe
B:Show that every two norms on finite dimension linear space are equivalent
C: Let f be a linear function from a normed space X in to a normed space Y, show that
continuous at x, E X iff for any sequence (x) in X converge to Xo then the sequence
(f(x)) converge to (f(x)) in Y.
Q3: A:Let M be a closed subspace of a normed space X, constract a linear space X/M as
normed space
B: Let A be a finite dimension subspace of a Banach space X, show that A is closed.
C: Show that every finite dimension normed space is Banach space.
• Plane II is spanned by the vectors:
P12
P2 = 1
• Subspace W is spanned by the vectors:
W₁ =
-- () ·
2
1
W2 =
0
Three streams - Stream A, Stream B, and Stream C - flow into a lake. The flow rates of these streams are
not yet known and thus to be found. The combined water inflow from the streams is 300 m³/h. The rate of
Stream A is three times the combined rates of Stream B and Stream C. The rate of Stream B is 50 m³/h less
than half of the difference between the rates of Stream A and Stream C.
Find the flow rates of the three streams by setting up an equation system Ax = b and solving it for x.
Provide the values of A and b. Assuming that you get to an upper-triangular matrix U using an elimination
matrix E such that U = E A, provide also the components of E.
Chapter 6 Solutions
Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY