Lecture 30: Linear transformations and their matrices
... First, we have to choose two bases, say v1 , v2 , ..., vn of Rn to give coordi nates to the input vectors and w1 , w2 , ..., wm of Rm to give coordinates to the output vectors. We want to find a matrix A so that T (v) = Av, where v and Av get their coordinates from these bases. The first column of A ...
... First, we have to choose two bases, say v1 , v2 , ..., vn of Rn to give coordi nates to the input vectors and w1 , w2 , ..., wm of Rm to give coordinates to the output vectors. We want to find a matrix A so that T (v) = Av, where v and Av get their coordinates from these bases. The first column of A ...
Linearly independence Definition: Consider a set of n
... Definition: The minimum number of linearly independent vector to span a space is the dimension of the vector space. In the above example, the vector space V 3 (R) has a dimension of 3, because that is the minimum of linearly independent vectors that is required to span it. Definition: Consider a ve ...
... Definition: The minimum number of linearly independent vector to span a space is the dimension of the vector space. In the above example, the vector space V 3 (R) has a dimension of 3, because that is the minimum of linearly independent vectors that is required to span it. Definition: Consider a ve ...
Final Exam Review
... 1. If sec θ = - in quadrant III, find the exact value of the cot θ. 2. Use the unit circle to find both exact values between 0 and 2 for 3. Use a sketch of a right triangle to find the exact value for ...
... 1. If sec θ = - in quadrant III, find the exact value of the cot θ. 2. Use the unit circle to find both exact values between 0 and 2 for 3. Use a sketch of a right triangle to find the exact value for ...
Physics Revision: Vectors and Scalars
... protractor (or set square) to make an angle of 900 and you then draw the 3 km line going east (to the right) at 900 to the 4 km and make it exactly 6cm long (1 km ≙ 2 cm; 3km ≙ 6cm). At the end of this line (at 6 cm) you have found the exact position of B. You now join A to B and measure the length ...
... protractor (or set square) to make an angle of 900 and you then draw the 3 km line going east (to the right) at 900 to the 4 km and make it exactly 6cm long (1 km ≙ 2 cm; 3km ≙ 6cm). At the end of this line (at 6 cm) you have found the exact position of B. You now join A to B and measure the length ...
1 Polarization of Light
... Classically the intensity of light is proportional to the square of the electric field so the outgoing intensity is proportional to the magnitude squared of the amplitude. In this example the intensity I = I0 sin2 θ. Question: One light beam is linearly polarized at 45 degrees; a second beam is Righ ...
... Classically the intensity of light is proportional to the square of the electric field so the outgoing intensity is proportional to the magnitude squared of the amplitude. In this example the intensity I = I0 sin2 θ. Question: One light beam is linearly polarized at 45 degrees; a second beam is Righ ...
Primer on Index Notation
... — the basis vectors — you cannot really understand how vectors are used in physics. Sometimes names other than {~ex , ~ey , ~ez } are given to the Cartesian basis vectors. Griffiths, for example, uses x̂, ŷ, ẑ, with the carets reminding us that these are unit vectors. However, in my opinion it is ...
... — the basis vectors — you cannot really understand how vectors are used in physics. Sometimes names other than {~ex , ~ey , ~ez } are given to the Cartesian basis vectors. Griffiths, for example, uses x̂, ŷ, ẑ, with the carets reminding us that these are unit vectors. However, in my opinion it is ...