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Math  /  Algebra

QuestionConsider the inner product on M22M_{22} defined by <U,V>=u1v1+u2v2+u3v3+u4v4<U, V>=u_{1} v_{1}+u_{2} v_{2}+u_{3} v_{3}+u_{4} v_{4} where U=[u1u2u3u4]U=\left[\begin{array}{ll}u_{1} & u_{2} \\ u_{3} & u_{4}\end{array}\right] and v=[v1v2v3v4]v=\left[\begin{array}{ll}v_{1} & v_{2} \\ v_{3} & v_{4}\end{array}\right] Using this inner product, the matrices [−22−4−4]\left[\begin{array}{cc}-2 & 2 \\ -4 & -4\end{array}\right] and [11−33]\left[\begin{array}{cc}1 & 1 \\ -3 & 3\end{array}\right] are orthogonal True False

Studdy Solution

STEP 1

1. The inner product is defined for matrices of size 2×22 \times 2.
2. Two matrices are orthogonal if their inner product is zero.

STEP 2

1. Identify the elements of the matrices.
2. Compute the inner product using the given definition.
3. Determine if the inner product is zero to check orthogonality.

STEP 3

Identify the elements of the matrices U U and V V .
For matrix U=[−22−4−4] U = \left[\begin{array}{cc}-2 & 2 \\ -4 & -4\end{array}\right] , we have: - u1=−2 u_1 = -2 - u2=2 u_2 = 2 - u3=−4 u_3 = -4 - u4=−4 u_4 = -4
For matrix V=[11−33] V = \left[\begin{array}{cc}1 & 1 \\ -3 & 3\end{array}\right] , we have: - v1=1 v_1 = 1 - v2=1 v_2 = 1 - v3=−3 v_3 = -3 - v4=3 v_4 = 3

STEP 4

Compute the inner product <U,V> <U, V> using the formula:
<U,V>=u1v1+u2v2+u3v3+u4v4 <U, V> = u_1 v_1 + u_2 v_2 + u_3 v_3 + u_4 v_4
Substitute the identified elements:
<U,V>=(−2)(1)+(2)(1)+(−4)(−3)+(−4)(3) <U, V> = (-2)(1) + (2)(1) + (-4)(-3) + (-4)(3)
Calculate each term:
=−2+2+12−12 = -2 + 2 + 12 - 12
Combine the terms:
=0 = 0

STEP 5

Determine if the matrices are orthogonal by checking if the inner product is zero.
Since <U,V>=0 <U, V> = 0 , the matrices are orthogonal.
The statement that the matrices are orthogonal is:
True \boxed{\text{True}}

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