MAT2611 ASSIGNMENT 5 2023
Problem 15
𝑝 = 3 − 𝑥 + 2𝑥 2
𝑆 = {𝑝1 , 𝑝2 , 𝑝3 }
𝑝1 = 3, 𝑝2 = 𝑥, 𝑝3 = 𝑥 2
𝑝 = 3 − 𝑥 + 2𝑥 2
𝑝 = 1 × 3 − 1 × 𝑥 + 2 × 𝑥2
𝑝 = 1𝑝1 − 1𝑝2 + 2𝑝3
The coordinate vector for 𝑝 relative to 𝑆 is:
(1, −1, 2)
Problem 16
Let 𝐴 be an 𝑚 × 𝑛 matrix.
Note that 𝐴 has 𝑚 rows and 𝑛 columns.
(a) The maximum possible rank of 𝐴.
The maximum possible rank of the matrix 𝐴 is 𝑚𝑖𝑛(𝑚, 𝑛).
The rank of a matrix is defined as the maximum number of linearly independent rows or
columns in the matrix. Since matrix 𝐴 has dimensions 𝑚 × 𝑛, it can have at most
𝑚𝑖𝑛(𝑚, 𝑛) linearly independent rows or columns.
If 𝑚 < 𝑛, then the maximum number of linearly independent rows in 𝐴 is 𝑚. In this case,
the remaining (𝑛 − 𝑚) rows can be expressed as linear combinations of the 𝑚 linearly
independent rows, so the rank of 𝐴 is 𝑚.
If 𝑚 > 𝑛, then the maximum number of linearly independent columns in 𝐴 is 𝑛. In this
case, the remaining (𝑚 − 𝑛) columns can be expressed as linear combinations of the 𝑛
linearly independent columns, so the rank of 𝐴 is 𝑛.
If 𝑚 = 𝑛, then the maximum number of linearly independent rows or columns in 𝐴 is
𝑚 = 𝑛. In this case, the rank of 𝐴 is equal to both 𝑚 and 𝑛.
Problem 15
𝑝 = 3 − 𝑥 + 2𝑥 2
𝑆 = {𝑝1 , 𝑝2 , 𝑝3 }
𝑝1 = 3, 𝑝2 = 𝑥, 𝑝3 = 𝑥 2
𝑝 = 3 − 𝑥 + 2𝑥 2
𝑝 = 1 × 3 − 1 × 𝑥 + 2 × 𝑥2
𝑝 = 1𝑝1 − 1𝑝2 + 2𝑝3
The coordinate vector for 𝑝 relative to 𝑆 is:
(1, −1, 2)
Problem 16
Let 𝐴 be an 𝑚 × 𝑛 matrix.
Note that 𝐴 has 𝑚 rows and 𝑛 columns.
(a) The maximum possible rank of 𝐴.
The maximum possible rank of the matrix 𝐴 is 𝑚𝑖𝑛(𝑚, 𝑛).
The rank of a matrix is defined as the maximum number of linearly independent rows or
columns in the matrix. Since matrix 𝐴 has dimensions 𝑚 × 𝑛, it can have at most
𝑚𝑖𝑛(𝑚, 𝑛) linearly independent rows or columns.
If 𝑚 < 𝑛, then the maximum number of linearly independent rows in 𝐴 is 𝑚. In this case,
the remaining (𝑛 − 𝑚) rows can be expressed as linear combinations of the 𝑚 linearly
independent rows, so the rank of 𝐴 is 𝑚.
If 𝑚 > 𝑛, then the maximum number of linearly independent columns in 𝐴 is 𝑛. In this
case, the remaining (𝑚 − 𝑛) columns can be expressed as linear combinations of the 𝑛
linearly independent columns, so the rank of 𝐴 is 𝑛.
If 𝑚 = 𝑛, then the maximum number of linearly independent rows or columns in 𝐴 is
𝑚 = 𝑛. In this case, the rank of 𝐴 is equal to both 𝑚 and 𝑛.
