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• Recall:

• Linear transformation T (associated with A w.r.t standard basis):

   T:R³ → R² is defined by

• Matrix equation (Equivalent to the system of linear equations): --Select an option-- X = B AX = B BX = A Submit

• Linear map equation (Equivalent to the system of linear equations): --Select an option-- T = b b(x) = T T(x) = b Submit

Reason

• Consistency of linear map equation:

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See the diagram below

• Summary:

Conditions for consistency:

• AX = B is consistent ⇔ Rank A = Rank A ⁺
• T(x) = b is consistent ⇔ b ∈ Range T
• Rank A = Rank A ⁺ ⇔ b ∈ Range T

Following are equivalent:

• System of linear equations is consistent
• AX = B is consistent
• T(x) = b is consistent