Take a matrix A = <vector r_1, vector r_2, ..., vector r_m>as row vectors. Let vector x be in the null space of A. Show that vector x is perpendicular to all of the vector r_k.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem provides the information that vector `bar x` is in the null space of matrix A, hence `bar x` checks the equation `A*bar x = 0` , such that:

`((bar r_1, bar r_2,....,bar r_k,..., bar r_n))*((barx_1),bar x_2),(...),(bar x_k),(...),(bar x_n)) = ((0,0,.....,0))`

Performing the matrices multiplication yields:

`((bar r_1*bar x_1 + bar r_2*bar x_2 + ... + bar r_k*bar x_k + ... + bar r_n*bar x_n)) = ((0,0,....,0))`

Equating corresponding members yields:

`bar r_1*bar x_1 + bar r_2*bar x_2 + ... + bar r_k*bar x_k + ... + bar r_n*bar x_n = 0`

Hence, the summation of scalar products `sum_(k=1)^n bar r_k*bar x_k = 0` if the vectors are perependicular, `bar r_k _|_ bar x_k.`

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