1 VECTOR SPACES AND SUBSPACES
... • A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . In general, all ten vector space axioms must be verified to show that a set W with addition and scalar multiplication forms a vector space. However, if ...
... • A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . In general, all ten vector space axioms must be verified to show that a set W with addition and scalar multiplication forms a vector space. However, if ...
R n
... Suppose TA : Rn Rn is a one-to-one linear operator The matrix A is invertible. TA-1 : Rn Rn is itself a linear operator; it is called the inverse of TA. TA(TA-1(x)) = AA-1x = Ix = x and TA-1(TA (x)) = A-1Ax = Ix = x TA ◦ TA-1 = TAA-1 = TI and TA-1 ◦ TA = TA-1A = TI If w is the image of x ...
... Suppose TA : Rn Rn is a one-to-one linear operator The matrix A is invertible. TA-1 : Rn Rn is itself a linear operator; it is called the inverse of TA. TA(TA-1(x)) = AA-1x = Ix = x and TA-1(TA (x)) = A-1Ax = Ix = x TA ◦ TA-1 = TAA-1 = TI and TA-1 ◦ TA = TA-1A = TI If w is the image of x ...
