Write x as the sum of two vectors, one in span{u1,u2,u3} and one in span {u4}. You may assume that {u1,u2,u3,u4} is an orthogonal set.
U1 U2 U3 U4 x
[0] [3] [1] [5] [10]
[1] [5] [0] [-3] [-8]
[-4] [1] [1] [-1] [2]
[-1] [1] [-4] [1] [0]
Any help or ideas on how to solve this would be much appreciated. Thanks!
U1 U2 U3 U4 x
[0] [3] [1] [5] [10]
[1] [5] [0] [-3] [-8]
[-4] [1] [1] [-1] [2]
[-1] [1] [-4] [1] [0]
Any help or ideas on how to solve this would be much appreciated. Thanks!
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Normalize the vectors (divide each vector by its length) in order to convert an orthogonal set to an orthonormal set.
In other words, calculate ui* = ui / (length of ui), for 1 <= i <= 4, to form the set {u1*,u2*,u3*,u4*}, which becomes an orthonormal basis for R^4.
Note that x is in R^4. Then x is the sum of the two vectors
(x dot u1*)u1* + (x dot u2*)u2* + (x dot u3*)u3*, which is in span {u1,u2,u3}
and
(x dot u4*)u4*, which is in span {u4}.
In other words, calculate ui* = ui / (length of ui), for 1 <= i <= 4, to form the set {u1*,u2*,u3*,u4*}, which becomes an orthonormal basis for R^4.
Note that x is in R^4. Then x is the sum of the two vectors
(x dot u1*)u1* + (x dot u2*)u2* + (x dot u3*)u3*, which is in span {u1,u2,u3}
and
(x dot u4*)u4*, which is in span {u4}.