For the V vector space contains all 2 times 2 matrices.

Nannie Mack

Nannie Mack

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For the V vector space contains all 2×2 matrices. Determine whether the T:VR1 is the linear transformation over the T(A)=a + 2b  c + d, where A=[abcd]

Answer & Explanation



Skilled2020-12-26Added 91 answers

For the vector space V, that contains all 2×2 matrices
Consider two matrices be lie in that vector space V be,
A=[abcd]B=[pqrs] and
Such that
T(A)=a + 2b  c + d
T(B)=p + 2q  r + s
Perform additional operation.
A + B=[abcd]+[pqrs]
T(A) + T(B)=(a + 2b  c + d) + (p + 2q  r + s)
=a + p + 2b + 2q  c  r + d + s
=(a + p) + 2(b + q)  (c + r) + (d + s)
Also, T(A + B)=(a + p) + 2(b + q)  (c + r) + (d + s)
Thus it shows that T(A+B)=T(A)+T(B), that means it satisfies the addition property
Consider the condition of
T(kA)=ka + 2kb  kc + kd
=k(a + 2b  c + d)
So the following expression also satisfies the scalar property.
As for the vector space V, the property of addition and scalar are satisfied. Thus the expression of T(A)=a+2bc+d is the linear transformation

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