Clearly state: True or False - If True, provide a concise explanation If False, provide an explicit counterexample - Let Ai = 6 be a consistent, inhomogeneous system of linear equations. A solution to the system is given by i = A- . Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly dependent. %3D
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![For each question:
- Clearly state: True or False
- If True, provide a concise explanation
- If False, provide an explicit counterexample
1. Let A = b be a consistent, inhomogeneous system of linear equations. A solution to the system
is given by i = A-
2. Let S, T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly
dependent.
3. Let S C P(R) be the set of all even degree polynomials. Then S forms a subspace of P(R).
4. Consider two maps S, T : R" R". The composition SoT is a linear map if and only if both
S and T are linear maps
5. Any invertible 3 x 3 real matrix M is a matrix of change of basis [C]g for some bases B and C.
6. Let W be a finite dimensional subspace of an inner product space V. For each w e W, there
exists a unique v EV such that w is an orthogonal projection of v on W.
7. Let A be a 4 x 4 matrix such that the homogeneous equation Af = õ has general solution
i = Xữ for some non-zero vector TER. Then, rank(A) = 3.
%3D
8. Let V be a inner product space and let a e V be a non-zero vector. The subset U defined by
U = {ie V(F, ā) = 0}
is a subspace of V
9. If A ER is a common eigenvalue of two n x n matrices A and B, then A is also an eigenvalue
of A+ B.
10. If a square matrix M has a stable distribution, then it must be regular](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b3a1bd4-9647-4d71-a211-bbafc7b1359c%2F901026cd-8f8a-4b10-bebc-adfdc0e395ec%2Fdori9r9_processed.jpeg&w=3840&q=75)
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