1. It is known that the n-dimensional vector A1A2... A (n-1) is linearly independent and the non-zero vector B is orthogonal to AI. It is proved that A1, A2, A3... A (n-1) and B are linearly independent. 2. The following vector groups are transformed into standard vector groups A1 = (1, - 1,1) TA2 = (- 1,1) TA3 = (1,1, - 1) T3 by Schmidt's standard orthogonalization method. Let a = (A1, a)

1. It is known that the n-dimensional vector A1A2... A (n-1) is linearly independent and the non-zero vector B is orthogonal to AI. It is proved that A1, A2, A3... A (n-1) and B are linearly independent. 2. The following vector groups are transformed into standard vector groups A1 = (1, - 1,1) TA2 = (- 1,1) TA3 = (1,1, - 1) T3 by Schmidt's standard orthogonalization method. Let a = (A1, a)

1.k1a1+k2a2+… +K (n-1) a (n-1) + KNB = 0, left times b transpose, because orthogonal, so B transpose times AI equals 0, so kn = 0, and because A1, A2 An-1 is linearly independent, so K1 = K2 = =B1 = A1, B2 = a2 - {(A2, B1) / (B1, B1)} * B1, B3 = a3 - {(A3, B2) / (b

Linear algebra proving problem, proving n-dimensional vector group α 1, α 2 The necessary and sufficient condition for linear independence of α n is that any n-dimensional vector α can be expressed linearly by them

It is proved that: 1) sufficiency obviously, because n + 1 n-dimensional vectors must be linearly related, so a can be determined by A1, A2 Because a is an arbitrary n-dimensional vector, a can be represented by A1, A2 The linear representation of an means A1, A2 If A1, A2 A linear correlation, a linear correlation

It is proved that in n-dimensional vector space, if α 1. α 2... α n is linearly independent, then any vector β can be linearly represented by α 1. α 2... α n

In n-dimensional vector space, any n + 1 vectors are linearly correlated, so α 1. α 2... α n, β are linearly correlated. Let C1 * α 1 + C2 * α 2... + CN * α n + C * β = 0 (where C1 Cn, C is not all 0) if C = 0, then α 1. α 2... α n is linearly correlated, contradictory! So C is not 0, we can know that β = - (C

Ask a linear algebra problem ~ n-dimensional vector group A1 = (1,0,0... 0) A2 = (1,1,0... 0) an = (1,1,... 1)
It is proved that vector group A1, A2... An is equivalent to n-dimensional unit vector group E1, E2... En

Firstly, A1 = E1, A2 = E1 + E2,..., an = E1 + E2 +... + en, so vector group A1, A2,..., an can be expressed linearly by E1, E2,..., en. Secondly, E1 = A1, E2 = a2-a1,..., en = an-a (n-1), so vector group E1, E2,..., en can be expressed linearly by A1, A2,..., an